Linear Map Inverses are Linear

Theorem

If a linear transformation $S:V\to W$ has an inverse $T:W\to V$, then $T$ is linear

Proof

• Assume that $S$ is linear and $T$ is its inverse
• Then, apply the linearity test for $T$
• Pick arbitrary $x,y\in W$
• Pick arbitrary $c\in \mathbb{F}$
• Then, consider $T(cx+y)$
• Note that since $S$ is Surjective, and since $x,y\in W$, $x,y$ can be rewritten in terms of $S$
• $x=S(v_1)$
• $y=S(v_2)$
• Then, we get $T(cS(v_1)+S(v_2))$
• Note that $S$ is linear, so we can apply linearity properties
• $=T(S(c{v}_1+v_2))$
• Note that the two are inverses, so they undo eachother
• $=c{v}_1+v_2$
• $=cT(x)+T(y)$ by definition of $T$
• Therefore, $T$ is a linear map by linearity test