Exercise 19

Define a projection to be a linear transformation $t\colon V\to V$ with the property that repeating the projection does nothing more than does the projection alone: $(t\mathbin{\circ}t)\,(\vec{v})=t(\vec{v})$ for all $\vec{v}\in V$.

• Show that orthogonal projection into a line has that property.

• Show that projection along a subspace has that property.

• Show that for any such $t$ there is a basis $B=\langle\vec{\beta}_{1},\ldots,\vec{\beta}_{n}\rangle$ for $V$ such that

  $$t(\vec{\beta}_{i})=\begin{cases}\vec{\beta}_{i}&i=1,2,\dots,\,r\\ \vec{0}&i=r+1,r+2,\dots,\,n\end{cases}$$

  where $r$ is the rank of $t$.

• Conclude that every projection is a projection along a subspace.

• Also conclude that every projection has a representation

  $${\rm Rep}_{B,B}(t)=\left(\begin{array}{c|c}I&Z\\ \hline\cr Z&Z\end{array}\right)$$

  in block partial-identity form.