Equality and Operations on Linear Transformations

Equality

If

$$L(\vec{x})=M(\vec{x})$$

for every $\vec{x}\in {\mathbb{R}}^n$, then the two transformations are equal ($L=M$).

• Basically if for all xs, if the outputs are equal, then the transformations are equal.

Operations

$$\begin{array}{r}(L+M)(\vec{x})=L(\vec{x})+M(\vec{x})\\ (cL)(\vec{x})=cL(\vec{x})\end{array}$$

• Works by summing the respective “entries”, like in a matrix.
• This can be done for the tf’s corresponding matrix too $[L]$.