Properties of Linear Transformations

Theorem: Transforming the Zero Vector

Every linear transformation $T:V\to W$ always transforms the zero vector $V$ to the zero vector of zero vector:

$$T(0_V)=0_W$$

PROOF

TODO

Theorem: Linearity of Composition

The composition $g\circ f:V\to W$ of two linear transformations $f:V\to U$ and $g:U\to W$ is also a linear transformation.

PROOF

$$\begin{array}{rl}g\circ f(\lambda \mathbf{u}+\mu \mathbf{v})& =g(f(\lambda \mathbf{u}+\mu \mathbf{v}))\\ & =g(\lambda f(\mathbf{u})+\mu f(\mathbf{v}))\\ & =g(\lambda f(\mathbf{u}))+g(\mu f(\mathbf{v}))\\ & =\lambda g(f(\mathbf{u}))+\mu g(f(\mathbf{v}))\\ & =\lambda g\circ f(\mathbf{u})+\mu g\circ f(\mathbf{v})\end{array}$$

Theorem: Linearity of the Inverse Transformation

If $T$ is a bijective linear transformation, then its inverse $T^{-1}$ is also a bijective linear transformation.

PROOF

TODO