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.

A concise guide to linear dependence and independence, and to bases and dimension of vector spaces: definitions, key propositions, replacement theorem, and subspace dimension.

Define vector spaces and subspaces with canonical examples (R^n, matrix, and function spaces). Focus on matrix spaces: symmetric/skew, triangular, and diagonal subspaces.

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.

Learn what vectors are, how to represent them, and the basics of vector operations (addition, scalar multiplication). Build intuition for linear combinations and span.

Overview of Web Vitals and Lighthouse scoring—what each metric means, how it’s measured, and target thresholds for LCP, INP, CLS, TBT, FCP, and Speed Index to improve performance.

Learn about the definition of gravitational field vectors and gravitational potential according to Newton's law of universal gravitation, and examine two important related examples: the shell theorem and galactic rotation curves.

Explore the method of undetermined coefficients, a powerful technique for solving specific nonhomogeneous linear ODEs with constant coefficients, widely used in engineering for models like vibrating systems and RLC circuits.

Explore the structure of the general solution for second-order nonhomogeneous linear ODEs in relation to their homogeneous counterparts. This post proves the existence of a general solution and the non-existence of singular 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.