91Ó°ÊÓ

Let \(V\) and \(W\) be vector spaces over the same field, and let \(T: V \rightarrow W\) be a linear transformation. For any \(H \in \mathcal{B}(\mathrm{W})\), define \(\widehat{\mathrm{T}}(H): \mathrm{V} \times \mathrm{V} \rightarrow F\) by \(\widehat{\mathbf{T}}(H)(x, y)=H(\mathbf{T}(x), \mathbf{T}(y))\) for all \(x, y \in \mathbf{V}\). Prove the following results. (a) If \(H \in \mathcal{B}(\mathrm{W})\), then \(\widehat{\mathrm{T}}(H) \in \mathcal{B}(\mathrm{V})\). (b) \(\widehat{\mathrm{T}}: \mathcal{B}(\mathrm{W}) \rightarrow \mathcal{B}(\mathrm{V})\) is a linear transformation. (c) If \(\mathrm{T}\) is an isomorphism, then so is \(\widehat{\mathrm{T}}\).

Let \(\mathrm{T}\) be a normal operator on a finite-dimensional complex inner product space \(\mathrm{V}\). Use the spectral decomposition \(\lambda_{1} \mathrm{~T}_{1}+\lambda_{2} \mathrm{~T}_{2}+\cdots+\lambda_{k} \mathrm{~T}_{k}\) of \(\mathrm{T}\) to prove the following results. (a) If \(g\) is a polynomial, then $$ g(\mathbf{T})=\sum_{i=1}^{k} g\left(\lambda_{i}\right) \mathbf{T}_{i} . $$ (b) If \(\mathrm{T}^{n}=\mathrm{T}_{0}\) for some \(n\), then \(\mathrm{T}=\mathrm{T}_{0}\). (c) Let \(U\) be a linear operator on \(V\). Then \(U\) commutes with \(T\) if and only if \(U\) commutes with each \(T_{i}\). (d) There exists a normal operator \(U\) on \(V\) such that \(U^{2}=T\). (e) \(\mathrm{T}\) is invertible if and only if \(\lambda_{i} \neq 0\) for \(1 \leq i \leq k\). (f) \(\mathrm{T}\) is a projection if and only if every eigenvalue of \(\mathrm{T}\) is 1 or 0 . (g) \(\mathrm{T}=-\mathrm{T}^{*}\) if and only if every \(\lambda_{i}\) is an imaginary number.

Let \(\mathrm{T}\) be a normal operator on a finite-dimensional complex inner product space \(\mathrm{V}\), and let \(\mathrm{W}\) be a subspace of \(\mathrm{V}\). Prove that if \(\mathrm{W}\) is T-invariant, then \(\mathrm{W}\) is also \(\mathrm{T}^{*}\)-invariant. Hint: Use Exercise 24 of Section \(5.4\).

Prove that if \(\lambda\) is an eigenvalue of \(A A^{*}\), then \(\lambda\) is an eigenvalue of \(A^{*} A\). This completes the proof of the lemma to Corollary 2 to Theorem \(6.44\).

For each of the given systems of linear equations, (i) If the system is consistent, find the unique solution having minimum norm. (ii) If the system is inconsistent, find the "best approximation to a solution" having minimum norm, as described in Theorem \(6.30\) (b). (Use your answers to parts (a) and (f) of Exercise 6.) \(x_{1}+x_{2}=1\) (a) \(x_{1}+x_{2}=2\) (b) \(x_{1}-2 x_{3}+x_{4}=-1\)

Use Corollary 1 of the spectral theorem to show that if \(\mathrm{T}\) is a normal operator on a complex finite-dimensional inner product space and \(U\) is a linear operator that commutes with \(\mathrm{T}\), then \(\mathrm{U}\) commutes with \(\mathrm{T}^{*}\).

Assume the notation of Theorem \(6.32\). (a) Prove that for any ordered basis \(\beta, \psi_{\beta}\) is linear. (b) Let \(\beta\) be an ordered basis for an \(n\)-dimensional space \(\mathrm{V}\) over \(F\), and let \(\phi_{\beta}: \mathrm{V} \rightarrow \mathrm{F}^{n}\) be the standard representation of \(\mathrm{V}\) with respect to \(\beta\). For \(A \in \mathbf{M}_{n \times n}(F)\), define \(H: \mathbf{V} \times \mathbf{V} \rightarrow F\) by \(H(x, y)=\) \(\left[\phi_{\beta}(x)\right]^{t} A\left[\phi_{\beta}(y)\right]\). Prove that \(H \in \mathcal{B}(\mathrm{V})\). Can you establish this as a corollary to Exercise 7? (c) Prove the converse of (b): Let \(H\) be a bilinear form on V. If \(A=\psi_{\beta}(H)\), then \(H(x, y)=\left[\phi_{\beta}(x)\right]^{t} A\left[\phi_{\beta}(y)\right]\).

Prove that if \(\left\\{w_{1}, w_{2}, \ldots, w_{n}\right\\}\) is an orthogonal set of nonzero vectors, then the vectors \(v_{1}, v_{2}, \ldots, v_{n}\) derived from the Gram-Schmidt process satisfy \(v_{i}=w_{i}\) for \(i=1,2, \ldots, n\). Hint: Use mathematical induction.

Provide reasons why each of the following is not an inner product on the given vector spaces. (a) \(\langle(a, b),(c, d)\rangle=a c-b d\) on \(\mathrm{R}^{2}\). (b) \(\langle A, B\rangle=\operatorname{tr}(A+B)\) on \(\mathrm{M}_{2 \times 2}(R)\). (c) \(\langle f(x), g(x)\rangle=\int_{0}^{1} f^{\prime}(t) g(t) d t\) on \(\mathrm{P}(R)\), where ' denotes differentiation.

Let \(\beta\) be a basis for a finite-dimensional inner product space. (a) Prove that if \(\langle x, z\rangle=0\) for all \(z \in \beta\), then \(x=0\). (b) Prove that if \(\langle x, z\rangle=\langle y, z\rangle\) for all \(z \in \beta\), then \(x=y\).