Show that a linear transformation is invertible. Let $V$ be a finite-dimensional inner product space, and let $W \subset V$ be a subspace.
Define $T : V \to V$ by $T(v) = v + \operatorname{Proj}_W (v)$.
Show that $T$ is invertible.
Help, please?
 A: To show that a map is invertible, you have a few options:


*

*Find an inverse (this might be hard here).

*In the finite dimensional case, show that the map is surjective (also might be hard here, but easier than finding an inverse map).

*In the finite dimensional case, show that the map is injective (this should be the right approach here).
One way to show that a map is injective (one-to-one) is to show that the kernel is trivial.  In other words, the only vector so that $T(v)=0$ is the zero vector.
Assume that $T(v)=0$, then $v+\operatorname{Proj}_Wv=0$.  Therefore, $-\operatorname{Proj}_Wv=v$, so $v$ is in $W$ and $\operatorname{Proj}_W(v)=v$.  Therefore, $T(v)=0$ means $2v=0$, so $v=0$.  
This proves the kernel is $0$, so $T$ is injective, so (since $V$ is finite dimensional), $T$ is an isomorphism.
A: $$T^{-1}(v)=\frac12\operatorname{Proj}_W(v) +\operatorname{Proj}_{W^{\top}}(v)$$
A: As noted by @Abishanka Saha, we can find the inverse $T^{-1}$ explicitly.
Let $z \in V$. We have to find $v \in V$ such that $z = T(v) = v + \operatorname{Proj}_W v$.
We have:
$$\underbrace{\operatorname{Proj}_W(z)}_{\in W} + \underbrace{\operatorname{Proj}_{W^\perp } (z)}_{\in W^\perp} = z = v + \operatorname{Proj}_W (v) = 2\operatorname{Proj}_W(v) + \operatorname{Proj}_{W^\perp} (v) = \underbrace{\operatorname{Proj}_W(2v)}_{\in W} + \underbrace{\operatorname{Proj}_{W^\perp}(v)}_{\in W^\perp}$$
Since $W \oplus W^\perp = V$, the decomposition of $z$ is unique, so we can equate the summands:
$$\operatorname{Proj}_W(z) = \operatorname{Proj}_{W}(2v) \implies \operatorname{Proj}_W(v) = \frac12 \operatorname{Proj}_W(z)$$
$$\operatorname{Proj}_{W^\perp}(z) = \operatorname{Proj}_{W^\perp}(v) $$
So we conclude $v = \operatorname{Proj}_W(v) + \operatorname{Proj}_{W^\perp}(v) = \frac12 \operatorname{Proj}_{W}(z) + \operatorname{Proj}_{W^\perp}(z)$.
Hence, we conclude that the inverse is given by $T^{-1}(z) = \frac12 \operatorname{Proj}_{W}(z) + \operatorname{Proj}_{W^\perp}(z)$ for $z \in V$.
