Theorem 32.1 - Properties of Matrix Transformations

Every matrix transformation must satisfy:

$$\begin{array}{r}{f}_A(\vec{x}+\vec{y})=f_A(\vec{x})+f_A(\vec{y})\\ \\ f_A(c\vec{x})=c{f}_A(\vec{x})\end{array}$$

• Proof in reference makes sense; it basically uses properties of matrix-vector product.

Matrix Transformations preserve vector sums and scalar multiplication–linear combinations. So now we define function which preserve linear combinations: Linear Transformations.