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\title{MAT 247 \\ Assignment 2 \\ Due Thursday January 27}



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\item (Axler 6.18) Let $ V, \langle, \rangle $ be an inner product space and let $ P : V \rightarrow V $ be a linear operator.  Suppose that $ P^2 = P $ and that $ ||Pv || \le || v|| $ for all $ v \in V $.  Prove that $ P $ is an orthogonal projection.

\item (Axler 6.26) Let $ V, \langle, \rangle $ be an inner product space and fix $ v \in V $.  Define a linear map $ T : V \rightarrow \mathbb{F} $ by $ T(w) = \langle v, w \rangle $.  Find $ T^*(a) $ for $ a \in \mathbb{F} $.

\item (Axler 6.28) Let $ V, \langle, \rangle $ be an inner product space, let $ T : V \rightarrow V $ be a linear operator and let $ \lambda \in \mathbb{F} $.  Prove that $ \lambda $ is an eigenvalue of $ T $ if and only if $ \bar{\lambda} $ is an eigenvalue of $ T^* $.

\item (Axler 6.32) Suppose that $ A $ is an $ n\times m $ real matrix.  Prove that the dimension of the span of the rows of $ A $ equals the dimension of the span of the columns of $ A $.
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