Linear Transformation

A Transformation is a transformation $T:{\mathbb{R}}^n\to {\mathbb{R}}^m$ satisfying

$$\begin{array}{c}T(\vec{u}+\vec{v})=T(\vec{u})+T(\vec{v})\\ T(c\vec{u})=cT(\vec{u})\end{array}$$

1-1

TLDR: 1-1 $⟺$ every column of T is a pivot column

• If there is at most one location in the codomain for every location in the domain
• 1-1 iff standard matrix has pivot in every column

Onto

TLDR: Onto $⟺$ every row of T is a pivot row A linear transformation $T:{\mathbb{R}}^n\to {\mathbb{R}}^m$ is onto if there exists a location in the codomain for every location in the domain onto iff the standard matrix has a pivot in every row The matrix A has columns which span ${\mathbb{R}}^m$. The matrix A has all pivotal rows.

1-1 and Onto

need square matrix if 1-1 then onto if Onto then 1-1