Theorem

If $T:\mathbf{R}^n\rightarrow\mathbf{R}^m$ is a linear function, then $T(\vec{0})=\vec{0}$.

Proof

\[\begin{align} & T(\vec{0})=T(\vec{0}+\vec{0})\\ \Rightarrow&T(\vec{0})=T(\vec{0})+T(\vec{0})\\ \Rightarrow&T(\vec{0})=\vec{0}\qquad\square \end{align}\]

You may also enjoy

Theorem 1

less than 1 minute read

Theorem If $x,y\in \mathbf{R}^2$ and one is not a multiple of the other, then every vector in $\mathbf{R}^2$ is a linear combination of $x$ and $y$.