Matrix Multiplication Lesson 3, column based viewpoint

In this video, I focus on a column-based interpretation of matrix multiplication. We begin by returning to a key fact from the previous lesson: when you multiply a matrix A by a standard basis vector ej​, the result is the j-th column of A. Using this, we explore how to think of Ax as a linear combination of the columns of A, where the weights are given by the coordinates of x.

We apply this perspective to products of the form AB, showing that the j-th column of the matrix product can be written as Abj, where bj is the j-th column of B. This leads us to the conclusion that the product AB can be viewed as applying A to each column of B, and we show that the column space of AB is a subset of the column space of A.

Toward the end of the video, we examine how several familiar geometric transformations in the plane, e.g. reflections, projections, and rotations, can be described using matrices. In each case, we use the action of the matrix on the standard basis vectors to determine the appropriate 2 by 2 matrix.

The lesson includes exercises throughout, which I hope helps you build intuition about matrix multiplication in preparation for linear algebra, applied mathematics, or data science.

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