Linear Map Inverses are Unique

Theorem

If function $S:V\to W$ has an inverse $T:W\to V$ then, $T$ is unique

Proof

• Suppose $T_1:W\to V$ and $T_2:W\to V$ are both inverses of $S\to V\to W$
• Pick any $w\in W$
• We know that $S:V\to W$ is surjective and so $S(v)=w$
• $T_1(W)=T_1(S(v))=V$
• $T_2(W)=T_2(S(v))=V$
• Therefore, $T_1(W)=T_2(W)=V$ thus the two maps are the same