91Ó°ÊÓ

Let \(\mathrm{T}\) be a normal operator on a finite-dimensional inner product space V. Prove that \(N(T)=N\left(T^{*}\right)\) and \(R(T)=R\left(T^{*}\right)\). Hint: Use Theorem \(6.15\) and Exercise 12 of Section \(6.3\).

Simultaneous diagonalization. Let \(U\) and \(T\) be normal operators on a finite- dimensional complex inner product space \(\mathrm{V}\) such that \(\mathrm{TU}=\mathrm{UT}\). Prove that there exists an orthonormal basis for \(V\) consisting of vectors that are eigenvectors of both \(\mathbf{T}\) and \(\mathbf{U}\). Hint: Use the hint of Exercise 14 of Section \(6.4\) along with Exercise 8 .

Let \(T\) be a linear operator on an inner product space \(V\). Prove that \(\|\mathrm{T}(x)\|=\|x\|\) for all \(x \in \mathrm{V}\) if and only if \(\langle\mathrm{T}(x), \mathrm{T}(y)\rangle=\langle x, y\rangle\) for all \(x, y \in \mathrm{V}\). Hint: Use Exercise 20 of Section 6.1.

Let \(\mathrm{V}\) be an inner product space, and suppose that \(x\) and \(y\) are orthogonal vectors in \(\mathrm{V}\). Prove that \(\|x+y\|^{2}=\|x\|^{2}+\|y\|^{2}\). Deduce the Pythagorean theorem in \(R^{2}\). Visit goo.gl/1iTZzC for a solution.

Let \(W\) be a finite-dimensional subspace of an inner product space \(V\). Prove that \(\mathrm{V}=\mathrm{W} \oplus \mathrm{W}^{\perp}\). Using the definition on page 76, prove that there exists a projection \(\mathrm{T}\) on \(\mathrm{W}\) along \(\mathrm{W}^{\perp}\) that satisfies \(\mathrm{N}(\mathrm{T})=\mathrm{W}^{\perp}\). In addition, prove that \(\|\mathrm{T}(x)\| \leq\|x\|\) for all \(x \in \mathrm{V}\). Hint: Use Theorem \(6.6\) and Exercise 10 of Section 6.1.

Find an orthogonal matrix whose first row is \(\left(\frac{1}{3}, \frac{2}{3}, \frac{2}{3}\right)\).

Prove the parallelogram law on an inner product space \(\mathbf{V} ;\) that is, show that $$ \|x+y\|^{2}+\|x-y\|^{2}=2\|x\|^{2}+2\|y\|^{2} \quad \text { for all } x, y \in \mathrm{V} . $$ What does this equation state about parallelograms in \(R^{2}\) ?

Assume that \(\mathrm{T}\) is a linear operator on a complex (not necessarily finitedimensional) inner product space \(\mathrm{V}\) with an adjoint \(\mathrm{T}^{*}\). Prove the following results. (a) If \(\mathrm{T}\) is self-adjoint, then \(\langle\mathrm{T}(x), x\rangle\) is real for all \(x \in \mathrm{V}\). (b) If \(\mathrm{T}\) satisfies \(\langle\mathrm{T}(x), x\rangle=0\) for all \(x \in \mathrm{V}\), then \(\mathrm{T}=\mathrm{T}_{0}\). Hint: Replace \(x\) by \(x+y\) and then by \(x+i y\), and expand the resulting inner products. (c) If \(\langle\mathrm{T}(x), x\rangle\) is real for all \(x \in \mathrm{V}\), then \(\mathrm{T}\) is self-adjoint.

For a linear operator \(T\) on an inner product space \(V\), prove that \(T^{*} T=\) \(\mathrm{T}_{0}\) implies \(\mathrm{T}=\mathrm{T}_{0}\). Is the same result true if we assume that \(\mathrm{TT}^{*}=\mathrm{T}_{0}\) ?

Prove that \(\operatorname{cond}(A)=1\) if and only if \(A\) is a scalar multiple of a unitary or orthogonal matrix.