## Systems of Linear Equations (and System Equivalency) [Video]

I begin with 2 by 2 and 3 by 3 linear systems of equations. I then discuss consistent and inconsistent systems of linear equations and work through several examples. The general system of linear equations is defined, and parametrizing a solution set of a linear system is demonstrated. I explain augmented matrices, row operations, and when two linear systems of equations are called equivalent.

## Canonical Forms and Jordan Blocks

Are you trying to understand what a Jordan basis is? In this article, I go over Jordan’s Theorem and the Jordan Canonical Form. I work through several examples.

## Invariant Subspaces and Generalized Eigenvectors

In this article, I cover invariant subspaces of a linear transformation. After that, I discuss using generalized eigenvectors for finding invariant subspaces.

## Diagonalization of a Matrix (with Examples)

Do you know the Diagonalization Theorem? In this article, I cover this theorem, two corollaries, and work through many examples. The method of diagonalization is illustrated in the following examples.

## Eigenvalues and Eigenvectors (Find and Use Them)

Are you looking for more information on eigenvalues and eigenvectors? In this article, I work through several examples and go over the characteristic equation, and then I go over even more examples. All theorems are provided with proof.

## The Determinant of a Matrix (Theory and Examples)

Okay, so there is the definition of a determinant, and there are ways of computing the determinant. The definition of determinant involves patterns and a signature, and this approach is useful in proofs. But to determine the determinant in practice, it’s helpful to have expansion by cofactor. In this article, I detail both.

## Inner Products and Orthonormal Bases

You have been familiar with the Cartesian plane for a while, and so you know about angles. How do we work with angles in a vector-space? In this article, I go over inner-product spaces and work through the Cauchy-Schwarz Theorem and the Triangular Theorem. I also discuss orthonormal bases.

## Gram-Schmidt Process and QR Factorization

In this article, I explain what the Gram-Schmidt Process is and what a QR Factorization is. I do this by working through several examples. Also, I include the commutative diagrams for the corresponding transformations.

## Orthogonal Matrix and Orthogonal Projection Matrix

In this article, I cover orthogonal transformations in detail. After that, I present orthogonal and transpose properties and orthogonal matrices. Towards the end, I examine the orthogonal projection matrix and provide many examples and exercises.

## Orthonormal Bases and Orthogonal Projections

In this article, I begin with the fact that orthonormal vectors are linearly independent and thus form a basis for the subspaces generated by them. After that, I explore the orthogonal projection and properties of the orthogonal complement. Towards the end, I detail the Pythagorean Theorem, the Cauchy-Schwarz Theorem, the Law of Cosines, and the Triangular Inequality Theorem.