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]]> Linear Transformations, Null Space, and Image2025-09-18T00:00:00+09:002025-09-18T00:00:00+09:00https://www.yunseo.kim/am/posts/linear-transformation-nullspace-and-image/Yunseo KimDefine linear transformations and study their null space (kernel) and image (range). Prove rank–nullity, relate injectivity/surjectivity to rank and nullity, and show how bases determine linear maps. Define linear transformations and study their null space (kernel) and image (range). Prove rank–nullity, relate injectivity/surjectivity to rank and nullity, and show how bases determine linear maps.
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A special class of functions that preserve the structure of vector spaces are called linear transformations. They are fundamental across pure and applied mathematics, social and natural sciences, and engineering.
Definition Let $\mathbb{V}$ and $\mathbb{W}$ be $F$-vector spaces. A function $T: \mathbb{V} \to \mathbb{W}$ is called a linear transformation from $\mathbb{V}$ to $\mathbb{W}$ if, for all $\mathbf{x}, \mathbf{y} \in \mathbb{V}$ and $c \in F$, the following hold:
When $T$ is a linear transformation, we also simply say that $T$ is linear. A linear transformation $T: \mathbb{V} \to \mathbb{W}$ satisfies the following four properties.
$T$ linear $\quad \Rightarrow \quad T(\mathbf{0}) = \mathbf{0}$
When proving that a function is linear, it is often convenient to use Property 2.
Linear algebra has wide and varied applications in geometry because many important geometric maps are linear. In particular, the three principal geometric transformations—rotation, reflection, and projection—are linear transformations.
Two linear transformations occur especially often:
Identity and zero transformations For $F$-vector spaces $\mathbb{V}, \mathbb{W}$:
Identity transformation: the function $I_\mathbb{V}: \mathbb{V} \to \mathbb{V}$ defined by $I_\mathbb{V}(\mathbf{x}) = \mathbf{x}$ for all $\mathbf{x} \in \mathbb{V}$
Zero transformation: the function $T_0: \mathbb{V} \to \mathbb{W}$ defined by $T_0(\mathbf{x}) = \mathbf{0}$ for all $\mathbf{x} \in \mathbb{V}$
Many other familiar operations are linear transformations.
e.g. For vector spaces $\mathbb{V}, \mathbb{W}$, the identity $I: \mathbb{V} \to \mathbb{V}$ and the zero map $T_0: \mathbb{V} \to \mathbb{W}$ satisfy:
$\mathrm{N}(I) = \{\mathbf{0}\}$
$\mathrm{R}(I) = \mathbb{V}$
$\mathrm{N}(T_0) = \mathbb{V}$
$\mathrm{R}(T_0) = \{\mathbf{0}\}$
A key point going forward is that the null space and the image of a linear transformation are subspaces of the corresponding vector spaces.
Theorem 1 For vector spaces $\mathbb{V}, \mathbb{W}$ and a linear transformation $T: \mathbb{V} \to \mathbb{W}$, the sets $\mathrm{N}(T)$ and $\mathrm{R}(T)$ are subspaces of $\mathbb{V}$ and $\mathbb{W}$, respectively.
Proof Denote the zero vectors of $\mathbb{V}$ and $\mathbb{W}$ by $\mathbf{0}_\mathbb{V}$ and $\mathbf{0}_\mathbb{W}$, respectively.
Since $T(\mathbf{0}_\mathbb{V}) = \mathbf{0}_\mathbb{W}$, we have $\mathbf{0}_\mathbb{V} \in \mathrm{N}(T)$. Moreover, for $\mathbf{x}, \mathbf{y} \in \mathrm{N}(T)$ and $c \in F$,
Similarly, $T(\mathbf{0}_\mathbb{V}) = \mathbf{0}_\mathbb{W}$ implies $\mathbf{0}_\mathbb{W} \in \mathrm{R}(T)$. For all $\mathbf{x}, \mathbf{y} \in \mathrm{R}(T)$ and $c \in F$ (there exist $\mathbf{v}, \mathbf{w} \in \mathbb{V}$ with $T(\mathbf{v}) = \mathbf{x}$ and $T(\mathbf{w}) = \mathbf{y}$), we have
Furthermore, given a basis $\beta = \{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n \}$ of $\mathbb{V}$, we can find a generating set of the image $\mathrm{R}(T)$ as follows.
Theorem 2 For vector spaces $\mathbb{V}, \mathbb{W}$, a linear transformation $T: \mathbb{V} \to \mathbb{W}$, and a basis $\beta = \{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n \}$ of $\mathbb{V}$, we have
$\therefore$ Since both contain each other, $\mathrm{R}(T) = \mathrm{span}({T(\mathbf{v}): \mathbf{v} \in \beta })$. $\blacksquare$
This theorem remains valid even when the basis $\beta$ is infinite.
Dimension theorem
Because the null space and image are especially important subspaces, we give special names to their dimensions.
For vector spaces $\mathbb{V}, \mathbb{W}$ and a linear transformation $T: \mathbb{V} \to \mathbb{W}$, assume $\mathrm{N}(T)$ and $\mathrm{R}(T)$ are finite-dimensional.
Nullity: the dimension of $\mathrm{N}(T)$, denoted $\mathrm{nullity}(T)$
Rank: the dimension of $\mathrm{R}(T)$, denoted $\mathrm{rank}(T)$
For a linear transformation, the larger the nullity, the smaller the rank, and vice versa.
Theorem 3: Dimension theorem For vector spaces $\mathbb{V}, \mathbb{W}$ and a linear transformation $T: \mathbb{V}\to \mathbb{W}$, if $\mathbb{V}$ is finite-dimensional, then
Let $\dim(\mathbb{V}) = n$ and $\mathrm{nullity}(T) = \dim(\mathrm{N}(T)) = k$, and let a basis of $\mathrm{N}(T)$ be $\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k \}$.
We now show that $S = \{T(\mathbf{v}_{k+1}), T(\mathbf{v}_{k+2}), \dots, T(\mathbf{v}_n) \}$ is a basis of $\mathrm{R}(T)$. First, for $1 \leq i \leq k$, $T(\mathbf{v}_i) = 0$, so by Theorem 2,
Thus $S$ is linearly independent and is a basis of $\mathrm{R}(T)$.
\[\therefore \mathrm{rank}(T) = n - k = \dim{\mathbb{V}} - \mathrm{nullity}(T). \blacksquare\]
Linear transformations and injections/surjections
For linear transformations, injectivity and surjectivity are closely tied to rank and nullity.
Theorem 4 For vector spaces $\mathbb{V}, \mathbb{W}$ and a linear transformation $T: \mathbb{V} \to \mathbb{W}$,
\[T \text{ is injective} \quad \Leftrightarrow \quad \mathrm{N}(T) = \{\mathbf{0}\}.\]
Theorem 5 If finite-dimensional vector spaces $\mathbb{V}, \mathbb{W}$ have the same dimension and $T: \mathbb{V} \to \mathbb{W}$ is linear, then the following four statements are equivalent.
These two theorems are useful when deciding whether a given linear transformation is injective or surjective.
For an infinite-dimensional vector space $\mathbb{V}$ and a linear transformation $T: \mathbb{V} \to \mathbb{V}$, injectivity and surjectivity are not equivalent.
If a linear transformation is injective, the following theorem can be useful in some cases for testing whether a subset of the domain is linearly independent.
Theorem 6 For vector spaces $\mathbb{V}, \mathbb{W}$, an injective linear transformation $T: \mathbb{V} \to \mathbb{W}$, and a subset $S \subseteq \mathbb{V}$,
\[S \text{ is linearly independent} \quad \Leftrightarrow \quad \{T(\mathbf{v}): \mathbf{v} \in S \} \text{ is linearly independent.}\]
Linear transformations and bases
A key feature of linear transformations is that their action is determined by their values on a basis.
Theorem 7 Let $\mathbb{V}, \mathbb{W}$ be $F$-vector spaces, let $\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n \}$ be a basis of $\mathbb{V}$, and let $\mathbf{w}_1, \mathbf{w}_2, \dots, \mathbf{w}_n \in \mathbb{W}$. Then there exists a unique linear transformation $T: \mathbb{V} \to \mathbb{W}$ such that
\[T(\mathbf{v}_i) = \mathbf{w}_i \quad (i = 1, 2, \dots, n).\]
Proof For $\mathbf{x} \in \mathbb{V}$, the representation
Corollary 7-1 Let $\mathbb{V}, \mathbb{W}$ be vector spaces and suppose $\mathbb{V}$ has a finite basis $\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n \}$. If two linear transformations $U, T: \mathbb{V} \to \mathbf{W}$ satisfy $U(\mathbf{v}_i) = T(\mathbf{v}_i)$ for $i = 1, 2, \dots, n$, then $U = T$. In other words, if two linear transformations agree on a basis, they are equal.
]]> Linear Dependence and Independence, Bases and Dimension2025-09-16T00:00:00+09:002025-10-25T21:21:39+09:00https://www.yunseo.kim/am/posts/linear-dependence-and-independence-basis-and-dimension/Yunseo KimA concise guide to linear dependence and independence, and to bases and dimension of vector spaces: definitions, key propositions, replacement theorem, and subspace dimension. A concise guide to linear dependence and independence, and to bases and dimension of vector spaces: definitions, key propositions, replacement theorem, and subspace dimension.
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Given a vector space $\mathbb{V}$ and a subspace $\mathbb{W}$, suppose we wish to find a minimal finite subset $S$ that spans $\mathbb{W}$.
Let $S = \{\mathbf{u}_1, \mathbf{u}_2, \mathbf{u}_3, \mathbf{u}_4 \}$ with $\mathrm{span}(S) = \mathbb{W}$. How can we decide whether there exists a proper subset of $S$ that still spans $\mathbb{W}$? This is equivalent to asking whether some vector in $S$ can be written as a linear combination of the others. For example, a necessary and sufficient condition for expressing $\mathbf{u}_4$ as a linear combination of the remaining three vectors is the existence of scalars $a_1, a_2, a_3$ satisfying
If some vector in $S$ is a linear combination of the others, then there exists a representation of the zero vector as a linear combination of elements of $S$ in which at least one among $a_1, a_2, a_3, a_4$ is nonzero. The converse is also true: if there is a nontrivial linear combination of vectors in $S$ that equals the zero vector (i.e., at least one of $a_1, a_2, a_3, a_4$ is nonzero), then some vector in $S$ is a linear combination of the others.
Generalizing this, we define linear dependence and linear independence as follows.
Definition For a subset $S$ of a vector space $\mathbb{V}$, if there exist finitely many distinct vectors $\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n \in S$ and scalars $a_1, a_2, \dots, a_n$, not all $0$, such that $a_1\mathbf{u}_1 + a_2\mathbf{u}_2 + \cdots + a_n\mathbf{u}_n = \mathbf{0}$, then the set $S$ (and those vectors) is called linearly dependent. Otherwise, it is called linearly independent.
For any vectors $\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n$, if $a_1 = a_2 = \cdots = a_n = 0$ then $a_1\mathbf{u}_1 + a_2\mathbf{u}_2 + \cdots + a_n\mathbf{u}_n = \mathbf{0}$; this is called the trivial representation of the zero vector.
The following three propositions about linearly independent sets hold in every vector space. In particular, Proposition 3 is very useful for testing whether a finite set is linearly independent.
Proposition 1: The empty set is linearly independent. A set must be nonempty to be linearly dependent.
Proposition 2: A set consisting of a single nonzero vector is linearly independent.
Proposition 3: A set is linearly independent if and only if the only way to express $\mathbf{0}$ as a linear combination of its vectors is the trivial one.
The following theorems are also important.
Theorem 1 If $\mathbb{V}$ is a vector space and $S_1 \subseteq S_2 \subseteq \mathbb{V}$, then $S_2$ is linearly dependent whenever $S_1$ is linearly dependent.
Corollary 1-1 If $\mathbb{V}$ is a vector space and $S_1 \subseteq S_2 \subseteq \mathbb{V}$, then $S_1$ is linearly independent whenever $S_2$ is linearly independent.
Theorem 2 Let $\mathbb{V}$ be a vector space and $S$ a linearly independent subset. For a vector $\mathbf{v} \in \mathbb{V}\setminus S$, $S \cup \{\mathbf{v}\}$ is linearly dependent if and only if $\mathbf{v} \in \mathrm{span}(S)$.
In other words, if no proper subset of $S$ spans the same space as $S$, then $S$ is linearly independent.
Bases and dimension
Basis
A spanning set $S$ of $\mathbb{W}$ that is linearly independent has a special property: every vector in $\mathbb{W}$ can be expressed as a linear combination of $S$, and that expression is unique (Theorem 3). Thus, we define a linearly independent spanning set of a vector space to be a basis.
Definition of a basis For a vector space $\mathbb{V}$ and a subset $\beta$, if $\beta$ is linearly independent and spans $\mathbb{V}$, then $\beta$ is called a basis of $\mathbb{V}$. In this case, the vectors in $\beta$ are said to form a basis of $\mathbb{V}$.
$\mathrm{span}(\emptyset) = \{\mathbf{0}\}$ and $\emptyset$ is linearly independent. Therefore, $\emptyset$ is a basis of the zero space.
In particular, the following distinguished basis of $F^n$ is called the standard basis of $F^n$.
Definition of the standard basis For the vector space $F^n$, consider
Then the set $\{\mathbf{e}_1, \mathbf{e}_2, \dots, \mathbf{e}_n \}$ is a basis of $F^n$, called the standard basis.
Theorem 3 Let $\mathbb{V}$ be a vector space and $\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n \in \mathbb{V}$ be distinct vectors. A necessary and sufficient condition for $\beta = \{\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n \}$ to be a basis of $\mathbb{V}$ is that every vector $\mathbf{v} \in \mathbb{V}$ can be expressed as a linear combination of vectors in $\beta$, and that this expression is unique. That is, there exist unique scalars $(a_1, a_2, \dots, a_n)$ such that
By Theorem 3, if the distinct vectors $\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n$ form a basis of a vector space $\mathbb{V}$, then within $\mathbb{V}$, a vector $\mathbf{v}$ uniquely determines the scalar $n$-tuple $(a_1, a_2, \dots, a_n)$, and conversely a scalar $n$-tuple uniquely determines the corresponding vector $\mathbf{v}$. We will revisit this when studying invertibility and isomorphisms; in this case, $\mathbb{V}$ and $F^n$ are essentially the same.
Theorem 4 If $S$ is a finite set with $\mathrm{span}(S) = \mathbb{V}$, then some subset of $S$ is a basis of $\mathbb{V}$. In particular, in this case every basis of $\mathbb{V}$ is finite.
Many vector spaces fall under the scope of Theorem 4, but not all do. A basis need not be finite.{: .prompt-tip }
Dimension
Theorem 5: Replacement theorem Let $G$ be a set of $n$ vectors with $\mathrm{span}(G) = \mathbb{V}$. If $L$ is a subset of $\mathbb{V}$ consisting of $m$ linearly independent vectors, then $m \le n$. Moreover, there exists a set $H \subseteq G$ with $n-m$ vectors such that $\mathrm{span}(L \cup H) = \mathbb{V}$.
From this we obtain two very important corollaries.
Corollary 5-1 of the replacement theorem If a vector space $\mathbb{V}$ has a finite basis, then every basis of $\mathbb{V}$ is finite and all bases have the same number of vectors.
Hence the number of vectors in a basis of $\mathbb{V}$ is an invariant, intrinsic property of $\mathbb{V}$, called its dimension.
Definition of dimension A vector space that has a finite basis is called finite-dimensional; in this case, the number $n$ of basis elements is the dimension of the vector space, denoted $\dim(\mathbb{V})$. A vector space that is not finite-dimensional is called infinite-dimensional.
$\dim(\{\mathbf{0}\}) = 0$
$\dim(F^n) = n$
$\dim(\mathcal{M}_{m \times n}(F)) = mn$
The dimension of a vector space depends on the underlying field.
Over the complex field $\mathbb{C}$, the complex numbers form a 1-dimensional vector space with basis $\{1\}$
Over the real field $\mathbb{R}$, the complex numbers form a 2-dimensional vector space with basis $\{1,i\}$
In a finite-dimensional vector space $\mathbb{V}$, any subset with more than $\dim(\mathbb{V})$ vectors can never be linearly independent.
Corollary 5-2 of the replacement theorem Let $\mathbb{V}$ be a vector space of dimension $n$.
Any finite spanning set of $\mathbb{V}$ has at least $n$ vectors, and any spanning set of $\mathbb{V}$ with exactly $n$ vectors is a basis.
Any linearly independent subset of $\mathbb{V}$ with exactly $n$ vectors is a basis of $\mathbb{V}$. 3. Any linearly independent subset of $\mathbb{V}$ can be extended to a basis. That is, if $L \subseteq \mathbb{V}$ is linearly independent, there exists a basis $\beta \supseteq L$ of $\mathbb{V}$.
Dimension of subspaces
Theorem 6 In a finite-dimensional vector space $\mathbb{V}$, every subspace $\mathbb{W}$ is finite-dimensional and satisfies $\dim(\mathbb{W}) \le \dim(\mathbb{V})$. In particular, if $\dim(\mathbb{W}) = \dim(\mathbb{V})$, then $\mathbb{V} = \mathbb{W}$.
Corollary 6-1 For a subspace $\mathbb{W}$ of a finite-dimensional vector space $\mathbb{V}$, any basis of $\mathbb{W}$ can be extended to a basis of $\mathbb{V}$.
By Theorem 6, the dimension of a subspace of $\mathbb{R}^3$ can be $0,1,2,$ or $3$.
0-dimensional: the zero space $\{\mathbf{0}\}$ containing only the origin ($\mathbf{0}$)
1-dimensional: a line through the origin ($\mathbf{0}$)
2-dimensional: a plane containing the origin ($\mathbf{0}$)
3-dimensional: the entire 3D Euclidean space
]]> Vector Spaces, Subspaces, and Matrices2025-09-13T00:00:00+09:002025-10-28T18:44:54+09:00https://www.yunseo.kim/am/posts/vector-spaces-subspaces-and-matrices/Yunseo KimDefine vector spaces and subspaces with canonical examples (R^n, matrix, and function spaces). Focus on matrix spaces: symmetric/skew, triangular, and diagonal subspaces. Define vector spaces and subspaces with canonical examples (R^n, matrix, and function spaces). Focus on matrix spaces: symmetric/skew, triangular, and diagonal subspaces.
* Mathematical equations and diagrams included in posts may not display properly when viewed with a feed reader.
TL;DR
Matrix
The entry of a matrix $A$ in the $i$-th row and $j$-th column is denoted $A_{ij}$ or $a_{ij}$
Diagonal entry: an entry $a_{ij}$ with $i=j$
The components $a_{i1}, a_{i2}, \dots, a_{in}$ are the $i$-th row of the matrix
Each row of a matrix can be regarded as a vector in $F^n$
Moreover, a row vector in $F^n$ can be viewed as another matrix of size $1 \times n$
The components $a_{1j}, a_{2j}, \dots, a_{mj}$ are the $j$-th column of the matrix
Each column of a matrix can be regarded as a vector in $F^m$
Moreover, a column vector in $F^m$ can be viewed as another matrix of size $m \times 1$
Zero matrix: a matrix all of whose entries are $0$, denoted by $O$
Square matrix: a matrix with the same number of rows and columns
For two $m \times n$ matrices $A, B$, if $A_{ij} = B_{ij}$ for all $1 \leq i \leq m$, $1 \leq j \leq n$ (i.e., every corresponding entry agrees), then the two matrices are defined to be equal ($A=B$)
Transpose (transpose matrix): for an $m \times n$ matrix $A$, the $n \times m$ matrix $A^T$ obtained by swapping rows and columns of $A$
Symmetric matrix: a square matrix $A$ with $A^T = A$
Skew-symmetric matrix: a square matrix $B$ with $B^T = -B$
Triangular matrix
Upper triangular matrix: a matrix whose entries below the diagonal are all $0$ (i.e., $i>j \Rightarrow A_{ij}=0$), usually denoted by $U$
Lower triangular matrix: a matrix whose entries above the diagonal are all $0$ (i.e., $i<j \Rightarrow A_{ij}=0$), usually denoted by $L$
Diagonal matrix: an $n \times n$ square matrix whose off-diagonal entries are all $0$ (i.e., $i \neq j \Rightarrow M_{ij}=0$), usually denoted by $D$
Representative vector spaces
The $n$-tuples $F^n$:
The set of all $n$-tuples with entries in a field $F$
Denoted $F^n$; an $F$-vector space
Matrix space:
The set of all $m \times n$ matrices with entries in a field $F$
Denoted $\mathcal{M}_{m \times n}(F)$; a vector space
Function space:
For a nonempty set $S$ over a field $F$, the set of all functions from $S$ to $F$
Denoted $\mathcal{F}(S,F)$; a vector space
Subspace
A subset $\mathbb{W}$ of an $F$-vector space $\mathbb{V}$ is called a subspace of $\mathbb{V}$ if it is an $F$-vector space under the same addition and scalar multiplication as defined on $\mathbb{V}$
For every vector space $\mathbb{V}$, both $\mathbb{V}$ itself and $\{0\}$ are subspaces; in particular, $\{0\}$ is called the zero subspace
If a subset of a vector space contains the zero vector and is closed under linear combinations (i.e., if $\mathrm{span}(\mathbb{W})=\mathbb{W}$), then it is a subspace
As briefly noted in Vectors and Linear Combinations, the definitions of vectors and vector spaces as algebraic structures are as follows.
Definition A vector space (or linear space) $\mathbb{V}$ over a field $F$ is a set equipped with two operations, sum and scalar multiplication, satisfying the following eight axioms. Elements of the field $F$ are called scalars, and elements of the vector space $\mathbb{V}$ are called vectors.
Sum: For $\mathbf{x}, \mathbf{y} \in \mathbb{V}$, there exists a unique element $\mathbf{x} + \mathbf{y} \in \mathbb{V}$. We call $\mathbf{x} + \mathbf{y}$ the sum of $\mathbf{x}$ and $\mathbf{y}$.
Scalar multiplication: For $a \in F$ and $\mathbf{x} \in \mathbb{V}$, there exists a unique element $a\mathbf{x} \in \mathbb{V}$. We call $a\mathbf{x}$ a scalar multiple of $\mathbf{x}$.
For all $\mathbf{x},\mathbf{y} \in \mathbb{V}$, $\mathbf{x} + \mathbf{y} = \mathbf{y} + \mathbf{x}$. (commutativity of addition)
For all $\mathbf{x},\mathbf{y},\mathbf{z} \in \mathbb{V}$, $(\mathbf{x}+\mathbf{y})+\mathbf{z} = \mathbf{x}+(\mathbf{y}+\mathbf{z})$. (associativity of addition)
There exists $\mathbf{0} \in \mathbb{V}$ such that $\mathbf{x} + \mathbf{0} = \mathbf{x}$ for all $\mathbf{x} \in \mathbb{V}$. (zero vector, additive identity)
For each $\mathbf{x} \in \mathbb{V}$, there exists $\mathbf{y} \in \mathbb{V}$ such that $\mathbf{x}+\mathbf{y}=\mathbf{0}$. (additive inverse)
For each $\mathbf{x} \in \mathbb{V}$, $1\mathbf{x} = \mathbf{x}$. (multiplicative identity)
For all $a,b \in F$ and $\mathbf{x} \in \mathbb{V}$, $(ab)\mathbf{x} = a(b\mathbf{x})$. (associativity of scalar multiplication)
For all $a \in F$ and $\mathbf{x},\mathbf{y} \in \mathbb{V}$, $a(\mathbf{x}+\mathbf{y}) = a\mathbf{x} + a\mathbf{y}$. (distributivity of scalar multiplication over vector addition)
For all $a,b \in F$ and $\mathbf{x},\mathbf{y} \in \mathbb{V}$, $(a+b)\mathbf{x} = a\mathbf{x} + b\mathbf{x}$. (distributivity of scalar multiplication over field addition)
Strictly speaking, one should write “the $F$-vector space $\mathbb{V}$,” but when discussing vector spaces the specific field is often not essential; thus, when there is no risk of confusion, we omit $F$ and simply write “the vector space $\mathbb{V}$.”
Matrix spaces
Row and column vectors
The set of all $n$-tuples with entries in a field $F$ is denoted $F^n$. For $u = (a_1, a_2, \dots, a_n) \in F^n$ and $v = (b_1, b_2, \dots, b_n) \in F^n$, defining addition and scalar multiplication by
\[\begin{align*} u + v &= (a_1+b_1, a_2+b_2, \dots, a_n+b_n), \\ cu &= (ca_1, ca_2, \dots, ca_n) \end{align*}\]
makes $F^n$ into an $F$-vector space.
Vectors in $F^n$ are usually written as column vectors rather than standalone row vectors $(a_1, a_2, \dots, a_n)$:
The entry of a matrix $A$ in the $i$-th row and $j$-th column is denoted $A_{ij}$ or $a_{ij}$.
Each $a_{ij}$ ($1 \leq i \leq m$, $1 \leq j \leq n$) belongs to $F$.
An entry $a_{ij}$ with $i=j$ is called a diagonal entry.
The components $a_{i1}, a_{i2}, \dots, a_{in}$ form the $i$-th row of the matrix. Each row can be regarded as a vector in $F^n$, and, furthermore, a row vector in $F^n$ can be viewed as another matrix of size $1 \times n$.
The components $a_{1j}, a_{2j}, \dots, a_{mj}$ form the $j$-th column of the matrix. Each column can be regarded as a vector in $F^m$, and, furthermore, a column vector in $F^m$ can be viewed as another matrix of size $m \times 1$.
An $m \times n$ matrix whose entries are all $0$ is called the zero matrix, denoted $O$.
A matrix with the same number of rows and columns is called a square matrix.
For two $m \times n$ matrices $A, B$, if $A_{ij} = B_{ij}$ for all $1 \leq i \leq m$, $1 \leq j \leq n$ (i.e., every corresponding entry agrees), we define the matrices to be equal ($A=B$).
The set of all $m \times n$ matrices with entries in $F$ is denoted $\mathcal{M}_{m \times n}(F)$. For $\mathbf{A},\mathbf{B} \in \mathcal{M}_{m \times n}(F)$ and $c \in F$, defining addition and scalar multiplication by
makes $\mathcal{M}_{m \times n}(F)$ a vector space, called a matrix space.
This naturally extends the operations defined on $F^n$ and $F^m$.
Function spaces
For a nonempty set $S$ over a field $F$, $\mathcal{F}(S,F)$ denotes the set of all functions from $S$ to $F$. For $f,g \in \mathcal{F}(S,F)$, we declare $f$ and $g$ equal ($f=g$) if $f(s) = g(s)$ for all $s \in S$.
For $f,g \in \mathcal{F}(S,F)$, $c \in F$, and $s \in S$, defining addition and scalar multiplication by
makes $\mathcal{F}(S,F)$ a vector space, called a function space.
Subspaces
Definition A subset $\mathbb{W}$ of an $F$-vector space $\mathbb{V}$ is called a subspace of $\mathbb{V}$ if it is an $F$-vector space under the same addition and scalar multiplication as those defined on $\mathbb{V}$.
For every vector space $\mathbb{V}$, both $\mathbb{V}$ itself and $\{0\}$ are subspaces; in particular, $\{0\}$ is called the zero subspace.
Whether a subset is a subspace can be checked using the following theorem.
Theorem 1 For a vector space $\mathbb{V}$ and a subset $\mathbb{W}$, $\mathbb{W}$ is a subspace of $\mathbb{V}$ if and only if the following three conditions hold (with the operations inherited from $\mathbb{V}$):
In short, if it contains the zero vector and is closed under linear combinations (i.e., if $\mathrm{span}(\mathbb{W})=\mathbb{W}$), then it is a subspace.
The following theorems also hold.
Theorem 2
For any subset $S$ of a vector space $\mathbb{V}$, the span $\mathrm{span}(S)$ is a subspace of $\mathbb{V}$ containing $S$.
\[S \subset \mathrm{span}(S) \leq \mathbb{V} \quad \forall\ S \subset \mathbb{V}.\]
Any subspace of $\mathbb{V}$ that contains $S$ must contain the span of $S$.
\[\mathbb{W}\supset \mathrm{span}(S) \quad \forall\ S \subset \mathbb{W} \leq \mathbb{V}.\]
Theorem 3 For subspaces of a vector space $\mathbb{V}$, the intersection of any collection of such subspaces is again a subspace of $\mathbb{V}$.
Transpose, symmetric, and skew-symmetric matrices
The transpose $A^T$ of an $m \times n$ matrix $A$ is the $n \times m$ matrix obtained by swapping the rows and columns of $A$:
A matrix $A$ with $A^T = A$ is called symmetric, and a matrix $B$ with $B^T = -B$ is called skew-symmetric. Symmetric and skew-symmetric matrices must be square.
Let $\mathbb{W}_1$ and $\mathbb{W}_2$ be the sets of all symmetric and all skew-symmetric matrices in $\mathcal{M}_{n \times n}(F)$, respectively. Then $\mathbb{W}_1$ and $\mathbb{W}_2$ are subspaces of $\mathcal{M}_{n \times n}(F)$; that is, they are closed under addition and scalar multiplication.
Triangular and diagonal matrices
These two classes of matrices are also particularly important.
First, we collectively call the following two types of matrices triangular matrices:
Upper triangular matrix: a matrix whose entries below the diagonal are all $0$ (i.e., $i>j \Rightarrow A_{ij}=0$), usually denoted by $U$
Lower triangular matrix: a matrix whose entries above the diagonal are all $0$ (i.e., $i<j \Rightarrow A_{ij}=0$), usually denoted by $L$
An $n \times n$ square matrix in which all off-diagonal entries are $0$—that is, $i \neq j \Rightarrow M_{ij}=0$—is called a diagonal matrix, usually denoted by $D$. A diagonal matrix is both upper and lower triangular.
The sets of upper triangular matrices, lower triangular matrices, and diagonal matrices are all subspaces of $\mathcal{M}_{m \times n}(F)$.
]]> Inner Product and Norm2025-09-10T00:00:00+09:002025-10-15T05:48:53+09:00https://www.yunseo.kim/am/posts/inner-product-and-norm/Yunseo KimDefine the inner product and the dot product, derive vector length/norm from them, and see how to compute the angle between vectors in R^n and general inner product spaces. Define the inner product and the dot product, derive vector length/norm from them, and see how to compute the angle between vectors in R^n and general inner product spaces.
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In a general $F$-vector space, the definition of an inner product is as follows.
Definition of the inner product and inner product space Consider an $F$-vector space $\mathbb{V}$. An inner product on $\mathbb{V}$, denoted $\langle \mathbf{x},\mathbf{y} \rangle$, is a function that assigns to each ordered pair of vectors $\mathbf{x}, \mathbf{y} \in \mathbb{V}$ a scalar in $F$ and satisfies the following:
For all $\mathbf{x},\mathbf{y},\mathbf{z} \in \mathbb{V}$ and all $c \in F$,
If $\mathbf{x} \neq \mathbf{0}$, then $\langle \mathbf{x}, \mathbf{x} \rangle$ is positive.
An $F$-vector space $\mathbb{V}$ equipped with an inner product is called an inner product space. In particular, when $F=\mathbb{C}$ it is a complex inner product space, and when $F=\mathbb{R}$ it is a real inner product space.
In particular, the following inner product is called the standard inner product. One can check that it satisfies all four axioms above.
Definition of the standard inner product For two vectors in $F^n$, $\mathbf{x}=(a_1, a_2, \dots, a_n)$ and $\mathbf{y}=(b_1, b_2, \dots, b_n)$, the standard inner product on $F^n$ is defined by
When $F=\mathbb{R}$, complex conjugation is trivial, so the standard inner product becomes $\sum_{i=1}^n a_i b_i$. In this special case we often write $\mathbf{x} \cdot \mathbf{y}$ instead of $\langle \mathbf{x}, \mathbf{y} \rangle$ and call it the dot product or scalar product.
Definition of the dot product/scalar product For $\mathbf{v}=(v_1, v_2, \dots, v_n)$ and $\mathbf{w}=(w_1, w_2, \dots, w_n)$ in $\mathbb{R}^n$, the dot product (or scalar product) is defined by
To avoid confusion, I will refer to it as the dot product whenever possible.
In Euclidean space, the inner product coincides with the dot product, so when the context is clear, the dot product is often simply called the inner product. Strictly speaking, however, an inner product is a more general notion that includes the dot product as a special case.
flowchart TD
A["Inner Product"] -->|includes| B["Standard Inner Product"]
B -->|"when F = R (real numbers)"| C["Dot/Scalar Product"]
%% inclusion notation
C -. included in .-> B
B -. included in .-> A
Length/Norm of a Vector
For a vector $\mathbf{v}=(v_1, v_2, \dots, v_n)$ in $\mathbb{R}^n$, the Euclidean length of $\mathbf{v}$ is defined via the dot product as
In a general inner product space, the norm satisfies the following fundamental properties.
Theorem Let $\mathbb{V}$ be an $F$-inner product space and let $\mathbf{x}, \mathbf{y} \in \mathbb{V}$ and $c \in F$. Then:
$\|c\mathbf{x}\| = |c| \cdot \|\mathbf{x}\|$
The following hold:
$\|\mathbf{x}\| = 0 \iff \mathbf{x}=\mathbf{0}$
$\|\mathbf{x}\| \geq 0 \ \forall \mathbf{x}$
Cauchy–Schwarz inequality: $| \langle \mathbf{x}, \mathbf{y} \rangle | \leq \|\mathbf{x}\| \cdot \|\mathbf{y}\|$ (with equality if and only if one of $\mathbf{x}$ and $\mathbf{y}$ is a scalar multiple of the other)
Triangle inequality: $\| \mathbf{x} + \mathbf{y} \| \leq \|\mathbf{x}\| + \|\mathbf{y}\|$ (with equality if and only if one is a scalar multiple of the other and they point in the same direction)
Angle Between Vectors and Unit Vectors
A vector of length $1$ is called a unit vector. For two vectors $\mathbf{v}=(v_1, v_2, \dots, v_n)$ and $\mathbf{w}=(w_1, w_2, \dots, w_n)$ in $\mathbb{R}^n$, we have $\mathbf{v} \cdot \mathbf{w} = \|\mathbf{v}\| \cdot \|\mathbf{w}\| \cos\theta$, from which the angle $\theta$ between $\mathbf{v}$ and $\mathbf{w}$ ($0 \leq \theta \leq \pi$) can be obtained:
Definition Let $\mathbb{V}$ be an inner product space. For vectors $\mathbf{x}, \mathbf{y} \in \mathbb{V}$, if $\langle \mathbf{x}, \mathbf{y} \rangle = 0$, then $\mathbf{x}$ and $\mathbf{y}$ are said to be orthogonal or perpendicular. Moreover,
For a subset $S \subset \mathbb{V}$, if any two distinct vectors in $S$ are orthogonal, then $S$ is called an orthogonal set.
A vector $\mathbf{x} \in \mathbb{V}$ with $\|\mathbf{x}\|=1$ is called a unit vector.
If a subset $S \subset \mathbb{V}$ is an orthogonal set consisting only of unit vectors, then $S$ is called an orthonormal set.
A set $S = { \mathbf{v}_1, \mathbf{v}_2, \dots }$ is orthonormal if and only if $\langle \mathbf{v}_i, \mathbf{v}_j \rangle = \delta_{ij}$. Multiplying a vector by a nonzero scalar does not affect orthogonality.
For any nonzero vector $\mathbf{x}$, the vector $\cfrac{\mathbf{x}}{\|\mathbf{x}\|}$ is a unit vector. Obtaining a unit vector by multiplying a nonzero vector by the reciprocal of its length is called normalizing.
]]> Vectors and Linear Combinations2025-09-07T00:00:00+09:002025-10-28T20:47:49+09:00https://www.yunseo.kim/am/posts/vectors-and-linear-combinations/Yunseo KimLearn what vectors are, how to represent them, and the basics of vector operations (addition, scalar multiplication). Build intuition for linear combinations and span. Learn what vectors are, how to represent them, and the basics of vector operations (addition, scalar multiplication). Build intuition for linear combinations and span.
* Mathematical equations and diagrams included in posts may not display properly when viewed with a feed reader.
TL;DR
Definition of a vector
Vector in the narrow sense (Euclidean vector): a physical quantity that has both magnitude and direction
Vector in the broad, linear-algebraic sense: an element of a vector space
Ways to represent vectors
Arrow representation: the vector’s magnitude is the length of the arrow, and its direction is the arrow’s direction. It is easy to visualize and intuitive, but it is difficult to represent higher-dimensional vectors (4D and above) or non-Euclidean vectors.
Component representation: place the tail of the vector at the origin of a coordinate space and express the vector by the coordinates of its head.
For finitely many vectors $\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n$ and scalars $a_1, a_2, \dots, a_n$, a vector $\mathbf{v}$ satisfying $\mathbf{v} = a_1\mathbf{u}_1 + a_2\mathbf{u}_2 + \cdots + a_n\mathbf{u}_n$ is called a linear combination of $\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n$.
The numbers $a_1, a_2, \dots, a_n$ are called the coefficients of this linear combination.
Span
For a nonempty subset $S$ of a vector space $\mathbb{V}$, the set of all linear combinations formed from vectors in $S$, denoted $\mathrm{span}(S)$.
By definition, $\mathrm{span}(\emptyset) = \{0\}$.
For a subset $S$ of a vector space $\mathbb{V}$, if $\mathrm{span}(S) = \mathbb{V}$, then $S$ is said to generate (or span) $\mathbb{V}$.
Prerequisites
Coordinate plane/coordinate space
Field
What is a vector?
Vector in the narrow sense: Euclidean vector
Many physical quantities such as force, velocity, and acceleration carry not only magnitude but also directional information. A physical quantity that has both magnitude and direction is called a vector.
The definition above is the one used in mechanics in physics and in high-school-level mathematics. A vector in this geometric sense—“the magnitude and direction of a directed line segment,” grounded in physical intuition—is more precisely called a Euclidean vector.
Vector in the broad sense: an element of a vector space
In linear algebra, vectors are defined more broadly than Euclidean vectors, as an abstract algebraic structure:
Definition A vector space (or linear space) $\mathbb{V}$ over a field $F$ is a set equipped with two operations, sum and scalar multiplication, satisfying the following eight axioms. Elements of the field $F$ are called scalars, and elements of the vector space $\mathbb{V}$ are called vectors.
Sum: For any $\mathbf{x}, \mathbf{y} \in \mathbb{V}$, there exists a unique element $\mathbf{x} + \mathbf{y} \in \mathbb{V}$. We call $\mathbf{x} + \mathbf{y}$ the sum of $\mathbf{x}$ and $\mathbf{y}$.
Scalar multiplication: For any $a \in F$ and $\mathbf{x} \in \mathbb{V}$, there exists a unique element $a\mathbf{x} \in \mathbb{V}$. In this case, $a\mathbf{x}$ is called the scalar multiple of $\mathbf{x}$.
For all $\mathbf{x},\mathbf{y} \in \mathbb{V}$, $\mathbf{x} + \mathbf{y} = \mathbf{y} + \mathbf{x}$. (commutativity of addition)
For all $\mathbf{x},\mathbf{y},\mathbf{z} \in \mathbb{V}$, $(\mathbf{x}+\mathbf{y})+\mathbf{z} = \mathbf{x}+(\mathbf{y}+\mathbf{z})$. (associativity of addition)
There exists $\mathbf{0} \in \mathbb{V}$ such that $\mathbf{x} + \mathbf{0} = \mathbf{x}$ for all $\mathbf{x} \in \mathbb{V}$. (zero vector, additive identity)
For each $\mathbf{x} \in \mathbb{V}$, there exists $\mathbf{y} \in \mathbb{V}$ such that $\mathbf{x} + \mathbf{y} = \mathbf{0}$. (additive inverse)
For each $\mathbf{x} \in \mathbb{V}$, $1\mathbf{x} = \mathbf{x}$. (multiplicative identity)
For all $a,b \in F$ and $\mathbf{x} \in \mathbb{V}$, $(ab)\mathbf{x} = a(b\mathbf{x})$. (associativity of scalar multiplication)
For all $a \in F$ and $\mathbf{x},\mathbf{y} \in \mathbb{V}$, $a(\mathbf{x}+\mathbf{y}) = a\mathbf{x} + a\mathbf{y}$. (distributivity of scalar multiplication over vector addition)
For all $a,b \in F$ and $\mathbf{x} \in \mathbb{V}$, $(a+b)\mathbf{x} = a\mathbf{x} + b\mathbf{x}$. (distributivity of scalar multiplication over field addition)
This definition of a vector in linear algebra encompasses a broader class than the previously mentioned Euclidean vector. You can verify that Euclidean vectors satisfy these eight properties.
The origin and development of vectors are closely tied to practical problems in physics—such as describing force, motion, rotation, and fields quantitatively. The concept was first introduced as Euclidean vectors to meet the physical need to mathematically express natural phenomena. Mathematics then generalized and systematized these physical ideas, establishing formal structures such as vector spaces, inner products, and exterior products, leading to today’s definition of vectors. In other words, vectors are concepts demanded by physics and formalized by mathematics—an interdisciplinary product developed through close interaction between the two communities, rather than a creation of pure mathematics alone.
The Euclidean vectors handled in classical mechanics can be expressed within a more general framework mathematically. Modern physics actively uses not only Euclidean vectors but also more abstract notions defined in mathematics—vector spaces, function spaces, etc.—and attaches physical meaning to them. Hence it is inappropriate to regard the two definitions of a vector as merely “the physical definition” and “the mathematical definition.”
We will defer a deeper dive into vector spaces and, for now, focus on Euclidean vectors—vectors in the narrow sense that admit geometric representation in coordinate spaces. Building intuition with Euclidean vectors first will be helpful when generalizing to other kinds of vectors later.
Ways to represent vectors
Arrow representation
This is the most common and most geometrically intuitive representation. The vector’s magnitude is represented by the length of an arrow, and its direction by the direction of the arrow.
While intuitive, this arrow representation has clear limitations for higher-dimensional vectors (4D and above). Moreover, we will eventually need to handle non-Euclidean vectors that are not easily depicted geometrically, so it is important to become comfortable with the component representation described next.
Component representation
Regardless of where a vector is located, if its magnitude and direction are the same, we consider it the same vector. Therefore, given a coordinate space, if we fix the tail of the vector at the origin of that coordinate space, then an $n$-dimensional vector corresponds to an arbitrary point in $n$-dimensional space, and we can represent the vector by the coordinates of its head. This is called the component representation of a vector.
\[(a_1, a_2, \cdots, a_n) \in \mathbb{R}^n \text{ or } \mathbb{C}^n\]
Just as calculus starts from numbers $x$ and functions $f(x)$, linear algebra starts from vectors $\mathbf{v}, \mathbf{w}, \dots$ and their linear combinations $c\mathbf{v} + d\mathbf{w} + \cdots$. Every linear combination of vectors is built from the two basic operations above, sum and scalar multiplication.
Given finitely many vectors $\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n$ and scalars $a_1, a_2, \dots, a_n$, a vector $\mathbf{v}$ satisfying
is called a linear combination of $\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n$. The numbers $a_1, a_2, \dots, a_n$ are the coefficients of this linear combination.
Why are linear combinations important? Consider the following situation: $n$ vectors in $m$-dimensional space form the $n$ columns of an $m \times n$ matrix.
Describe all possible linear combinations $Ax = x_1\mathbf{v}_1 + x_2\mathbf{v}_2 + \cdots + x_n\mathbf{v}_n$. What do they form?
Given a desired output vector $b$, find numbers $x_1, x_2, \dots, x_n$ such that $Ax = b$.
We will return to the second question later; for now, focus on the first. To simplify, consider the case of two nonzero 2D vectors ($m=2$, $n=2$).
The linear combination $c\mathbf{v} + d\mathbf{w}$
A vector $\mathbf{v}$ in 2D has two components. For any scalar $c$, the vector $c\mathbf{v}$ traces an infinitely long line through the origin in the $xy$-plane, parallel to the original vector $\mathbf{v}$.
If the given second vector $\mathbf{w}$ is not on this line (i.e., $\mathbf{v}$ and $\mathbf{w}$ are not parallel), then $d\mathbf{w}$ traces another line. Combining these two lines, we see that the linear combination $c\mathbf{v} + d\mathbf{w}$ fills a single plane that includes the origin.
In this way, linear combinations of vectors form a vector space, a process called spanning.
Definition For a nonempty subset $S$ of a vector space $\mathbb{V}$, the set of all linear combinations formed from vectors in $S$ is called the span of $S$ and is denoted by $\mathrm{span}(S)$. By definition, $\mathrm{span}(\emptyset) = \{0\}$.
Definition For a subset $S$ of a vector space $\mathbb{V}$, if $\mathrm{span}(S) = \mathbb{V}$, then $S$ is said to generate (or span) $\mathbb{V}$.
Although we have not yet introduced concepts such as subspaces and bases, recalling this example will help you understand the concept of a vector space.
]]> Summary of Kaggle 'Pandas' Course (2) - Lessons 4–62025-08-24T00:00:00+09:002025-08-24T00:00:00+09:00https://www.yunseo.kim/am/posts/summary-of-kaggle-pandas-course-2/Yunseo KimPractical Pandas for data cleaning and wrangling: a concise summary of Kaggle’s free 'Pandas' course with added notes. This part covers Lessons 4–6—grouping/sorting, data types & missing values, renaming and combining. Practical Pandas for data cleaning and wrangling: a concise summary of Kaggle’s free 'Pandas' course with added notes. This part covers Lessons 4–6—grouping/sorting, data types & missing values, renaming and combining.
* Mathematical equations and diagrams included in posts may not display properly when viewed with a feed reader.
I summarize here what I studied through Kaggle’s Pandas course. Since it’s fairly long, I split it into two parts.
Group the reviews DataFrame by identical values in the taster_name column
Within each group, select the taster_name column
Return a Series with the count of non-missing values
In other words, the value_counts() method is essentially shorthand for the behavior above. Beyond count(), you can use any summary function similarly. For instance, to find the minimum price per score in the wine data:
Here len refers to Python’s built-in len(). In this example it reports the number of price (price) entries per group (country), including missing values. Since it accepts a DataFrame or Series as input, it can be used this way.
In contrast, pandas’ count() returns the count of non-missing values only.
This note isn’t in the original Kaggle course; I added it based on the official Python and pandas documentation.
MultiIndex
When you perform groupby-based transformations and analyses, you’ll sometimes get a DataFrame with a MultiIndex composed of more than one level.
That said, the method you’ll likely use most often is reset_index() to flatten back to a regular Index:
class="highlight">
1
countries_reviewed.reset_index()
country
province
len
0
Argentina
Mendoza Province
3264
1
Argentina
Other
536
…
…
…
…
423
Uruguay
San Jose
3
424
Uruguay
Uruguay
24
Sorting
Looking at countries_reviewed, you’ll notice grouped results are returned in index order. That is, the row order of a groupby result is determined by index values, not by data content.
When needed, you can sort explicitly using sort_values(). For example, to sort country–province pairs in ascending order by the number of entries (‘len’):
In practice, data rarely comes perfectly clean. More often than not, column types aren’t what you want and need conversion, and missing values appear throughout and must be handled carefully. For most data workflows, this stage is the biggest hurdle.
Data types
The data type of a DataFrame column or a Series is its dtype. Use the dtype attribute to check the type of a specific column. For example, to inspect the dtype of the price column in reviews:
class="highlight">
1
reviews.price.dtype
class="highlight">
1
dtype('float64')
Or use the dtypes attribute to inspect all column dtypes at once:
Pandas also supports “extension” dtypes such as categorical and various time-series types.
Missing values
Empty entries are represented as NaN (short for “Not a Number”). For technical reasons, NaN is always of dtype float64.
Pandas provides helper functions for missing data. We briefly saw something similar before: in addition to methods, pandas has standalone functions pd.isna and pd.notna. They return a single boolean or a boolean array indicating whether entries are missing (or not), and can be used like this:
class="highlight">
1
reviews[pd.isna(reviews.country)]
Often you’ll want to detect missing values and then fill them with appropriate replacements. One strategy is to use fillna() to replace NaNs with a chosen value. For example, replace all NaN in the region_2 column with "Unknown":
class="highlight">
1
reviews.region_2.fillna("Unknown")
Alternatively, you can use forward fill or backward fill to propagate the nearest valid value from above or below, via ffill() and bfill(), respectively.
Previously you could pass 'ffill'/'bfill' to the method parameter of fillna(), but this became deprecated starting in pandas 2.1.0. Prefer ffill() or bfill() directly instead.
Sometimes you need to replace a value with another even if it’s not missing. The original Kaggle course gives an example of a reviewer changing their Twitter handle. That’s a fine example, but here’s one that may feel more relatable to Korean readers:
Suppose South Korea split the northern part of Gyeonggi-do and established a new administrative region called Gyeonggibuk-do, and you have a dataset reflecting that change. Now imagine someone floated the harebrained idea of renaming Gyeonggibuk-do to Pyeonghwanuri Special Self-Governing Province, and actually managed to ram it through—a purely hypothetical scenario, of course. It’s scary how close something like this might have come to happening. You would then need to replace "Gyeonggibuk-do" with a new value like "Pyeonghwanuri State" or "Pyeonghwanuri Special Self-Governing Province" in the dataset. One way to do this in pandas is with replace():
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1
rok_2030_census.province.replace("Gyeonggibuk-do","Pyeonghwanuri Special Self-Governing Province")
With this snippet, you can effectively bulk-replace every "Gyeonggibuk-do" string in the province column of the rok_2030_census dataset with ‘that long one’. It’s a relief no one actually had to run code like this in real life.
String replacement is also useful during cleaning, since missingness is often encoded as strings like "Unknown", "Undisclosed", or "Invalid" rather than NaN. In real-world workflows such as OCR-ing old official documents into datasets, this may be the norm rather than the exception.
Lesson 6. Renaming and Combining
Sometimes you need to rename specific columns or index labels in a dataset. You’ll also frequently have to combine multiple DataFrames or Series.
Renaming
Use rename() to rename columns or index labels. It supports various input formats, but a Python dictionary is usually the most convenient. The following examples rename the points column to score and relabel index entries 0 and 1 to firstEntry and secondEntry:
In practice, renaming columns is common, while renaming index values is rare; for that purpose, it’s usually more convenient to use set_index()as we saw earlier.
Both the row and column axes have a name attribute. You can rename these axis names with rename_axis(). For example, label the row axis as wines and the column axis as fields:
You’ll often need to combine DataFrames or Series. Pandas provides three core tools for this, from simplest to most flexible: concat(), join(), and merge(). The Kaggle course focuses on the first two, noting that most merge() tasks can be done more simply with join().
concat() is the simplest: it stitches multiple DataFrames or Series along a given axis. It’s handy when the objects share the same fields (columns). By default, it concatenates along the index axis; specify axis=1 or axis='columns' to concatenate along columns.
According to the pandas docs, when building a DataFrame from many rows, avoid appending rows one by one in a loop. Instead, collect the rows in a list and perform a single concat().
join() is more complex: it attaches another DataFrame to a base DataFrame by aligning on the index. If the two DataFrames have overlapping column names, you must specify lsuffix and rsuffix to disambiguate them.
]]> The Wronskian, Existence and Uniqueness of Solutions2025-04-06T00:00:00+09:002025-07-11T21:22:11+09:00https://www.yunseo.kim/am/posts/wronskian-existence-and-uniqueness-of-solutions/Yunseo KimExplore the existence and uniqueness of solutions for second-order homogeneous linear ODEs with continuous variable coefficients. Learn to use the Wronskian to test for linear independence and see why these equations always have a general solution that encompasses all possible solutions. Explore the existence and uniqueness of solutions for second-order homogeneous linear ODEs with continuous variable coefficients. Learn to use the Wronskian to test for linear independence and see why these equations always have a general solution that encompasses all possible solutions.
* Mathematical equations and diagrams included in posts may not display properly when viewed with a feed reader.
TL;DR
For a second-order homogeneous linear ordinary differential equation with continuous variable coefficients $p$ and $q$ on an interval $I$
\[y^{\prime\prime} + p(x)y^{\prime} + q(x)y = 0\]
and initial conditions
\[y(x_0)=K_0, \qquad y^{\prime}(x_0)=K_1\]
the following four theorems hold.
Existence and Uniqueness Theorem for Initial Value Problems: The initial value problem consisting of the given equation and initial conditions has a unique solution $y(x)$ on the interval $I$.
Test for Linear Dependence/Independence using the Wronskian: For two solutions $y_1$ and $y_2$ of the equation, if there exists an $x_0$ in the interval $I$ where the Wronskian $W(y_1, y_2) = y_1y_2^{\prime} - y_2y_1^{\prime}$ is $0$, then the two solutions are linearly dependent. Furthermore, if there exists an $x_1$ in the interval $I$ where $W\neq 0$, then the two solutions are linearly independent.
Existence of a General Solution: The given equation has a general solution on the interval $I$.
Nonexistence of Singular Solutions: This general solution includes all solutions of the equation (i.e., no singular solutions exist).
Homogeneous Linear ODEs with Continuous Variable Coefficients
Previously, we examined the general solutions of Homogeneous Linear ODEs with Constant Coefficients and the Euler-Cauchy Equation. In this article, we extend the discussion to a more general case: a second-order homogeneous linear ordinary differential equation with arbitrary continuous variable coefficients $p$ and $q$.
We will investigate the existence and form of the general solution for this equation. Additionally, we will explore the uniqueness of the solution to the Initial Value Problem composed of the ODE ($\ref{eqn:homogeneous_linear_ode_with_var_coefficients}$) and the following two initial conditions:
To state the conclusion upfront, the core of this discussion is that a linear ordinary differential equation with continuous coefficients does not have a singular solution (a solution that cannot be obtained from the general solution).
Existence and Uniqueness Theorem for Initial Value Problems
Existence and Uniqueness Theorem for Initial Value Problems If $p(x)$ and $q(x)$ are continuous functions on some open interval $I$, and $x_0$ is in $I$, then the initial value problem consisting of Eqs. ($\ref{eqn:homogeneous_linear_ode_with_var_coefficients}$) and ($\ref{eqn:initial_conditions}$) has a unique solution $y(x)$ on the interval $I$.
The proof of existence will not be covered here; we will only look at the proof of uniqueness. Proving uniqueness is typically simpler than proving existence. If you are not interested in the proof, you may skip this section and proceed to Linear Dependence and Independence of Solutions.
Proof of Uniqueness
Let’s assume that the initial value problem consisting of the ODE ($\ref{eqn:homogeneous_linear_ode_with_var_coefficients}$) and initial conditions ($\ref{eqn:initial_conditions}$) has two solutions, $y_1(x)$ and $y_2(x)$, on the interval $I$. If we can show that their difference
\[y(x) = y_1(x) - y_2(x)\]
is identically zero on the interval $I$, this implies that $y_1 \equiv y_2$ on $I$, which means the solution is unique.
Since Eq. ($\ref{eqn:homogeneous_linear_ode_with_var_coefficients}$) is a homogeneous linear ODE, the linear combination $y$ of $y_1$ and $y_2$ is also a solution to the equation on $I$. Since $y_1$ and $y_2$ satisfy the same initial conditions ($\ref{eqn:initial_conditions}$), $y$ satisfies the conditions
This means that on the interval $I$, $F_1 z$ is non-increasing and $F_2 z$ is non-decreasing. By Eq. ($\ref{eqn:initial_conditions_*}$), we have $z(x_0) = 0$, so
Finally, dividing both sides of the inequalities by the positive functions $F_1$ and $F_2$, we can show the uniqueness of the solution as follows:
\[(z \leq 0) \ \& \ (z \geq 0) \quad \forall x \in I\] \[z = y^2 + {y^{\prime}}^2 = 0 \quad \forall x \in I\] \[\therefore y \equiv y_1 - y_2 \equiv 0 \quad \forall x \in I. \ \blacksquare\]
Linear Dependence and Independence of Solutions
Let’s briefly recall what we covered in Second-Order Homogeneous Linear ODEs. The general solution on an open interval $I$ is constructed from a basis $y_1$, $y_2$ on $I$, which is a pair of linearly independent solutions. Here, $y_1$ and $y_2$ being linearly independent on an interval $I$ means that for all $x$ in the interval, the following holds:
\[k_1y_1(x) + k_2y_2(x) = 0 \Leftrightarrow k_1=0\text{ and }k_2=0 \label{eqn:linearly_independent}\tag{8}\]
If the above is not satisfied, and $k_1y_1(x) + k_2y_2(x) = 0$ holds for at least one non-zero $k_1$ or $k_2$, then $y_1$ and $y_2$ are linearly dependent on the interval $I$. In this case, for all $x$ in the interval $I$,
Now let’s look at the following test for linear independence/dependence of solutions.
Test for Linear Dependence/Independence using the Wronskian i. If the ODE ($\ref{eqn:homogeneous_linear_ode_with_var_coefficients}$) has continuous coefficients $p(x)$ and $q(x)$ on an open interval $I$, then a necessary and sufficient condition for two solutions $y_1$ and $y_2$ of Eq. ($\ref{eqn:homogeneous_linear_ode_with_var_coefficients}$) to be linearly dependent on $I$ is that their Wronski determinant, or simply Wronskian, which is the following determinant,
In other words, if there exists an $x_1$ in the interval $I$ such that $W\neq 0$, then $y_1$ and $y_2$ are linearly independent on that interval $I$.
\[\begin{align*} \exists x_1 \in I: W(x_1)\neq 0 &\implies \forall x \in I: W(x)\neq 0 \\ &\implies y_1 \text{ and } y_2 \text{ are linearly independent} \end{align*}\]
The Wronskian was first introduced by the Polish mathematician Józef Maria Hoene-Wroński and was named after him posthumously in 11882 HE by the Scottish mathematician Sir Thomas Muir.
Proof
i. (a)
Let $y_1$ and $y_2$ be linearly dependent on the interval $I$. Then, either Eq. ($\ref{eqn:linearly_dependent}$a) or ($\ref{eqn:linearly_dependent}$b) holds on $I$. If Eq. ($\ref{eqn:linearly_dependent}$a) holds, then
Thus, we can confirm that the Wronskian $W(y_1, y_2)=0$ for all $x$ in the interval $I$.
i. (b)
Conversely, suppose that $W(y_1, y_2)=0$ for some $x = x_0$. We will show that $y_1$ and $y_2$ are linearly dependent on the interval $I$. Consider the system of linear equations for the unknowns $k_1$, $k_2$:
and the determinant of this matrix is $W(y_1(x_0), y_2(x_0))$. Since $\det(A) = W=0$, $A$ is a singular matrix that does not have an inverse matrix. Therefore, the system of equations ($\ref{eqn:linear_system}$) has a non-trivial solution $(c_1, c_2)$ other than the zero vector $(0,0)$, where at least one of $k_1$ and $k_2$ is not zero. Now, let’s introduce the function
\[y(x) = c_1y_1(x) + c_2y_2(x)\]
Since Eq. ($\ref{eqn:homogeneous_linear_ode_with_var_coefficients}$) is homogeneous and linear, by the Superposition Principle, this function is a solution of ($\ref{eqn:homogeneous_linear_ode_with_var_coefficients}$) on the interval $I$. From Eq. ($\ref{eqn:linear_system}$), we can see that this solution satisfies the initial conditions $y(x_0)=0$, $y^{\prime}(x_0)=0$.
Meanwhile, there exists a trivial solution $y^* \equiv 0$ that satisfies the same initial conditions $y^*(x_0)=0$, ${y^*}^{\prime}(x_0)=0$. Since the coefficients $p$ and $q$ of Eq. ($\ref{eqn:homogeneous_linear_ode_with_var_coefficients}$) are continuous, the uniqueness of the solution is guaranteed by the Existence and Uniqueness Theorem for Initial Value Problems. Therefore, $y \equiv y^*$. That is, on the interval $I$,
\[c_1y_1 + c_2y_2 \equiv 0\]
Since at least one of $c_1$ and $c_2$ is not zero, this does not satisfy ($\ref{eqn:linearly_independent}$), which means that $y_1$ and $y_2$ are linearly dependent on the interval $I$.
ii.
If $W(x_0)=0$ at some point $x_0$ in the interval $I$, then by i.(b), $y_1$ and $y_2$ are linearly dependent on the interval $I$. Then, by i.(a), $W\equiv 0$. Therefore, if there is even one point $x_1$ in the interval $I$ where $W(x_1)\neq 0$, then $y_1$ and $y_2$ are linearly independent. $\blacksquare$
The General Solution Includes All Solutions
Existence of a General Solution
If $p(x)$ and $q(x)$ are continuous on an open interval $I$, then the equation ($\ref{eqn:homogeneous_linear_ode_with_var_coefficients}$) has a general solution on the interval $I$.
Therefore, by the Test for Linear Dependence/Independence using the Wronskian, $y_1$ and $y_2$ are linearly independent on the interval $I$. Thus, these two solutions form a basis of solutions for Eq. ($\ref{eqn:homogeneous_linear_ode_with_var_coefficients}$) on the interval $I$, and a general solution $y = c_1y_1 + c_2y_2$ with arbitrary constants $c_1$, $c_2$ must exist on the interval $I$. $\blacksquare$
Nonexistence of Singular Solutions
If the ODE ($\ref{eqn:homogeneous_linear_ode_with_var_coefficients}$) has continuous coefficients $p(x)$ and $q(x)$ on some open interval $I$, then every solution $y=Y(x)$ of Eq. ($\ref{eqn:homogeneous_linear_ode_with_var_coefficients}$) on the interval $I$ is of the form
where $y_1$, $y_2$ form a basis of solutions for Eq. ($\ref{eqn:homogeneous_linear_ode_with_var_coefficients}$) on the interval $I$, and $C_1$, $C_2$ are suitable constants. That is, Eq. ($\ref{eqn:homogeneous_linear_ode_with_var_coefficients}$) does not have a singular solution, which is a solution that cannot be obtained from the general solution.
Proof
Let $y=Y(x)$ be any solution of Eq. ($\ref{eqn:homogeneous_linear_ode_with_var_coefficients}$) on the interval $I$. Now, by the Existence of a General Solution theorem, the ODE ($\ref{eqn:homogeneous_linear_ode_with_var_coefficients}$) has a general solution on the interval $I$:
Now we must show that for any $Y(x)$, there exist constants $c_1$, $c_2$ such that $y(x)=Y(x)$ on the interval $I$. Let’s first show that we can find values for $c_1$, $c_2$ such that for an arbitrary $x_0$ in $I$, we have $y(x_0)=Y(x_0)$ and $y^{\prime}(x_0)=Y^{\prime}(x_0)$. From Eq. ($\ref{eqn:general_solution}$), we get
Since $y_1$ and $y_2$ form a basis, the determinant of the coefficient matrix, which is the Wronskian $W(y_1(x_0), y_2(x_0))$, is non-zero. Therefore, Eq. ($\ref{eqn:vector_equation_2}$) can be solved for $c_1$ and $c_2$. Let the solution be $(c_1, c_2) = (C_1, C_2)$. Substituting this into Eq. ($\ref{eqn:general_solution}$) gives the following particular solution:
\[y^*(x) = C_1y_1(x) + C_2y_2(x).\]
Since $C_1$, $C_2$ are the solution to Eq. ($\ref{eqn:vector_equation_2}$),
]]> Sequences and Series2025-03-16T00:00:00+09:002025-05-13T16:24:07+09:00https://www.yunseo.kim/am/posts/sequences-and-series/Yunseo KimWe examine fundamental concepts of calculus such as the definition of sequences and series, convergence and divergence of sequences, convergence and divergence of series, and the definition of e, the base of natural logarithm. We examine fundamental concepts of calculus such as the definition of sequences and series, convergence and divergence of sequences, convergence and divergence of series, and the definition of e, the base of natural logarithm.
* Mathematical equations and diagrams included in posts may not display properly when viewed with a feed reader.
Sequences
In calculus, a sequence primarily refers to an infinite sequence. That is, a sequence is a function defined on the set of all natural numbers
\[\mathbb{N} := \{1,2,3,\dots\}\]
If the values of this function are real numbers, we call it a ‘real sequence’; if complex numbers, a ‘complex sequence’; if points, a ‘point sequence’; if matrices, a ‘matrix sequence’; if functions, a ‘function sequence’; if sets, a ‘set sequence’. However, all of these can be simply referred to as ‘sequences’.
Usually, for the field of real numbers $\mathbb{R}$, in a sequence $\mathbf{a}: \mathbb{N} \to \mathbb{R}$, we denote
*In the process of defining a sequence, instead of using the set of all natural numbers $\mathbb{N}$ as the domain, we can also use the set of non-negative integers
For example, when dealing with power series theory, it’s more natural to have $\mathbb{N}_0$ as the domain.
Convergence and Divergence
If a sequence $(a_n)$ converges to a real number $l$, we write
\[\lim_{n\to \infty} a_n = l\]
and call $l$ the limit of the sequence $(a_n)$.
The rigorous definition using the epsilon-delta argument is as follows:
\[\lim_{n\to \infty} a_n = l \overset{def}\Longleftrightarrow \forall \epsilon > 0,\, \exists N \in \mathbb{N}\ (n > N \Rightarrow |a_n - l| < \epsilon)\]
In other words, if for any positive $\epsilon$, there always exists a natural number $N$ such that $|a_n - l | < \epsilon$ when $n>N$, it means that the difference between $a_n$ and $l$ becomes infinitely small for sufficiently large $n$. Therefore, we define that a sequence $(a_n)$ satisfying this condition converges to the real number $l$.
A sequence that does not converge is said to diverge. The convergence or divergence of a sequence does not change even if a finite number of its terms are altered.
If each term of the sequence $(a_n)$ grows infinitely large, we write
\[\lim_{n\to \infty} a_n = \infty\]
and say that it diverges to positive infinity. Similarly, if each term of the sequence $(a_n)$ becomes infinitely small, we write
\[\lim_{n\to \infty} a_n = -\infty\]
and say that it diverges to negative infinity.
Basic Properties of Convergent Sequences
If sequences $(a_n)$ and $(b_n)$ both converge (i.e., have limits), then the sequences $(a_n + b_n)$ and $(a_n \cdot b_n)$ also converge, and
This is considered one of the most important constants in mathematics.
The term ‘natural constant’ is widely used only in Korea, but this is not a standard term. The official term registered in the mathematics terminology dictionary by the Korean Mathematical Society is ‘base of natural logarithm’, and the expression ‘natural constant’ cannot be found in this dictionary. Even in the Standard Korean Language Dictionary of the National Institute of Korean Language, the word ‘natural constant’ cannot be found, and in the dictionary definition of ‘natural logarithm’, it only mentions “a specific number usually denoted as e”. In English-speaking countries and Japan, there is no corresponding term, and in English, it’s mainly referred to as ‘the base of the natural logarithm’ or shortened to ‘natural base’, or ‘Euler’s number’ or ‘the number $e$’. Since the origin is unclear and it has never been recognized as an official term by the Korean Mathematical Society, and it’s not used anywhere else in the world except Korea, there’s no reason to insist on using such a term. Therefore, from now on, I will refer to it as ‘the base of natural logarithm’ or simply denote it as $e$.
The limit value $l$ is called the sum of the series $\sum a_n$. The symbol
\[\sum a_n\]
can represent either the series or the sum of the series, depending on the context.
A series that does not converge is said to diverge.
Basic Properties of Convergent Series
From the basic properties of convergent sequences, we obtain the following basic properties of convergent series. For a real number $t$ and two convergent series $\sum a_n$, $\sum b_n$,
The convergence of a series is not affected by changes in a finite number of terms. That is, if $a_n=b_n$ for all but finitely many $n$ in two sequences $(a_n)$, $(b_n)$, the series $\sum a_n$ converges if and only if the series $\sum b_n$ converges.
ከቀደሙት ደረጃዎች በኋላ bundle exec jekyll serve ትዕዛዝን በማስኬድ build ሙከራ ሲደረግ፣ 'relative_url_regex': target of repeat operator is not specified የሚል ስህተት ተፈጥሮ የbuild ሂደቱ አልተሳካም።
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Jekyll 4.3.4 Please append `--trace` to the `serve`command
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