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.
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.
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.