Matrices

Outside of the vector, the matrix is the most fundamental building block of linear algebra.

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📄️ Linear Transformations

Linear transformations are the bread and butter of linear algebra.

📄️ Dimensionality

Previously, we introduced the concept of linear transformations.

📄️ Composite Transformations

In the previous sections, we laid the groundwork for a visual understanding of transformations.

📄️ Determinant

In this section, we will introduce the concept of the determinant of a matrix.

📄️ Formalizing Linear Transformations

In the previous sections, we introduced the concept of linear transformations and how they can be represented by matrices.

Overview

Just like the previous section on vectors, this is a short part of the notes that provides a basic conceptual introduction to matrices.

We will begin with a motivating example - linear transformations, and then use it to define matrix-vector multiplication. (Usually it's done the other way around.) We will visualize linear transformations in both 2D and 3D space, and show that all linear transformations can be represented by matrices.

Then, we will discuss how matrices can be combined together with matrix multiplication. We will also explore some important properties of matrices, such as the determinant and the inverse.

After that, we will touch on some more abstract topics, such as the rank, null space, column space, and more.