Projections and adjoints

#linear_algebra

Let $V$ be a vector space and $T : V \to V$ be a linear transformation.
(a) Prove that if $T$ is a projection (i.e., $T^{2} = T$ ), then $V$ can be decomposed into the internal direct sum $V = null (T) \oplus range (T)$ .
(b) Suppose $V$ is an inner product space and $T^{*}$ is the adjoint of $T$ with respect to the inner product. Show that $null (T^{*})$ is the orthogonal complement of $range (T)$ .
(c) 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.

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L A T E X

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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}

Important note

Sometime after this exam was given, the exam syllabus was updated and the topic of general inner-product spaces was removed. As such, this problem does not appear in the problem bank.