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\title{MAT 247 \\ Assignment 3 \\ Due Thursday February 3}



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\item (Axler 6.29) Let $ V $ be an inner product space and let $ T : V \rightarrow V $ be a linear map.  Let $ U $ be a subspace of $ V $.  Show that $ U $ is invariant under $ T $ if and only if $ U^\perp $ is invariant under $ T^* $.

\item (Axler 7.2) Prove or give a counterexample: the product of any two self-adjoint operators on a finite-dimensional inner product space is self-adjoint.

\item (Axler 7.8) Show that there is no self-adjoint operator $ T : \mathbb{R}^3 \rightarrow \mathbb{R}^3 $ such that $ T(1,2,3) = 0 $ and $ T(2,5,7) = (2,5,7)$.

\item Consider the linear operator $ T : \mathbb{R}^2 \rightarrow \mathbb{R}^2 $ given by the matrix $ \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}. $  Show that $ T $ is self-adjoint.  Find an orthonormal basis for $ \mathbb{R}^2 $ consisting of eigenvectors for $ T $.
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