Endomorphism/Orthogonal sum/Adjoint/Exercise

Let ${}V$ be a finite-dimensional ${}{\mathbb {K} }$-vector space, endowed with inner product, and let

${}V=V_{1}\oplus V_{2}\,$

be a direct sum of the linear subspaces ${}V_{1}$ and ${}V_{2}$. Let

$\varphi _{1}\colon V_{1}\longrightarrow V_{1}$

and

$\varphi _{2}\colon V_{2}\longrightarrow V_{2}$

be linear mappings, and let

${}\varphi =\varphi _{1}\oplus \varphi _{2}\,$

denote their sum.

a) Suppose that the sum decomposition is orthogonal; that is, ${}V_{1}$ and ${}V_{2}$ are orthogonal to each other. Show

${}{\hat {\varphi }}={\hat {\varphi _{1}}}\oplus {\hat {\varphi _{2}}}\,.$

b) Show that the statement in part (a) does not hold when the sum decomposition is not orthogonal.

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