Projections and adjoints
Let
- Prove that if
is a projection (i.e., ), then can be decomposed into the internal direct sum . - Suppose
is an inner product space and is the adjoint of with respect to the inner product. Show that is the orthogonal complement of . - Suppose
is an inner product space and is an orthogonal projection, i.e., a projection for which the null space and range are orthogonal. Show that is self adjoint.
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Let $V$ be a vector space and $T:V\to V$ be a linear transformation.
\begin{enumerate}[label=\alph*)]
\item Prove that if $T$ is a projection (i.e., $T^2=T$), then $V$ can be decomposed into the internal direct sum $V=\operatorname{null}(T)\oplus \operatorname{range}(T)$.
\item Suppose $V$ is an inner product space and $T^*$ is the adjoint of $T$ with respect to the inner product. Show that $\operatorname{null}(T^*)$ is the orthogonal complement of $\operatorname{range}(T)$.
\item Suppose $V$ is an inner product space and $T$ is an orthogonal projection, i.e., a projection for which the null space and range are orthogonal. Show that $T$ is self adjoint.
\end{enumerate}