Linear Algebra Course by Kimberly Brehm

Sobre este curso

Linear algebra is one of the fastest ways to level up your problem-solving skills in school math and unlock tools used in science, engineering, data analysis, and economics. In this free online course, you will learn to think in a clear, structured way about vectors and matrices, turning complicated-looking problems into step-by-step procedures you can actually trust.

You will start by making sense of systems of linear equations and the logic behind row operations. Instead of memorizing tricks, you will understand what each transformation does to a system, how augmented matrices encode information, and how echelon forms reveal whether a system has one solution, infinitely many, or no solution at all. Along the way, you will get comfortable interpreting solution sets, identifying free variables, and connecting algebraic steps to meaning.

As the course progresses, you will move naturally from equations to vector thinking: expressing relationships as vector equations, building linear combinations, and using the matrix equation Ax=b to model real situations. This foundation leads into core matrix operations, invertibility, and the practical reasons inverses matter when solving problems efficiently. Determinants and their properties add another powerful lens, giving you reliable tests and insights about matrices beyond simple computation.

You will also develop a deeper conceptual understanding through vector spaces, subspaces, bases, dimension, rank, and the relationships among row space, column space, and null space. These ideas bring structure to what can feel like scattered techniques, helping you recognize patterns and choose methods with confidence. Later topics such as eigenvalues and eigenvectors, inner products, orthogonality, projections, and least squares show how linear algebra supports geometry, optimization, and best-fit modeling.

With embedded practice questions, you can check your understanding as you go and build the habits needed for exams and further study. By the end, you will not only perform calculations—you will understand what they mean and when to use them, leaving you better prepared for advanced mathematics and real-world applications.