Set of Linear Transformations is Isomorphic to Matrix Space/Motivation

Motivation for Set of Linear Transformations is Isomorphic to Matrix Space

What Set of Linear Transformations is Isomorphic to Matrix Space tells us is two things:

1. That the relative matrix of a linear transformation can be considered to be the same thing as the transformation itself
2. To determine the relative matrix for the composite of two linear transformations, what you do is multiply the relative matrices of those linear transformations.

Thus one has a means of direct arithmetical manipulation of linear transformations, thereby transforming geometry into algebra.

In fact, matrix multiplication was purposely defined (some would say designed) so as to produce exactly this result.

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 29$. Matrices