91Ó°ÊÓ

Label the following statements as true or false. Assume that the underlying inner product spaces are finite-dimensional. (a) All projections are self-adjoint. (b) An orthogonal projection is uniquely determined by its range. (c) Every self-adjoint operator is a linear combination of orthogonal projections. (d) If \(\mathrm{T}\) is a projection on \(\mathrm{W}\), then \(\mathrm{T}(x)\) is the vector in \(\mathrm{W}\) that is closest to \(x\). (e) Every orthogonal projection is a unitary operator.

In each part, apply the Gram-Schmidt process to the given subset \(S\) of the inner product space \(\mathrm{V}\) to obtain an orthogonal basis for \(\operatorname{span}(S)\). Then normalize the vectors in this basis to obtain an orthonormal basis \(\beta\) for \(\operatorname{span}(S)\), and compute the Fourier coefficients of the given vector relative to \(\beta\). Finally, use Theorem \(6.5\) to verify your result. (a) \(\mathrm{V}=\mathrm{R}^{3}, S=\\{(1,0,1),(0,1,1),(1,3,3)\\}\), and \(x=(1,1,2)\) (b) \(\mathbf{V}=\mathbf{R}^{3}, S=\\{(1,1,1),(0,1,1),(0,0,1)\\}\), and \(x=(1,0,1)\) (c) \(\mathrm{V}=\mathrm{P}_{2}(R)\) with the inner product \(\langle f(x), g(x)\rangle=\int_{0}^{1} f(t) g(t) d t\), \(S=\left\\{1, x, x^{2}\right\\}\), and \(h(x)=1+x\) (d) \(\mathrm{V}=\operatorname{span}(S)\), where \(S=\\{(1, i, 0),(1-i, 2,4 i)\\}\), and \(x=(3+i, 4 i,-4)\) (e) \(\mathrm{V}=\mathrm{R}^{4}, S=\\{(2,-1,-2,4),(-2,1,-5,5),(-1,3,7,11)\\}\), and \(x=\) \((-11,8,-4,18)\) (f) \(\mathbf{V}=\mathbf{R}^{4}, S=\\{(1,-2,-1,3),(3,6,3,-1),(1,4,2,8)\\}\), and \(x=(-1,2,1,1)\) (g) \(\mathrm{V}=\mathrm{M}_{2 \times 2}(R), S=\left\\{\left(\begin{array}{rr}3 & 5 \\ -1 & 1\end{array}\right),\left(\begin{array}{rr}-1 & 9 \\ 5 & -1\end{array}\right),\left(\begin{array}{rr}7 & -17 \\ 2 & -6\end{array}\right)\right\\}\), and \(A=\left(\begin{array}{rr}-1 & 27 \\ -4 & 8\end{array}\right)\) (h) \(\mathrm{V}=\mathrm{M}_{2 \times 2}(R), S=\left\\{\left(\begin{array}{ll}2 & 2 \\ 2 & 1\end{array}\right),\left(\begin{array}{rr}11 & 4 \\ 2 & 5\end{array}\right),\left(\begin{array}{ll}4 & -12 \\ 3 & -16\end{array}\right)\right\\}\), and \(A=\) \(\left(\begin{array}{rr}8 & 6 \\ 25 & -13\end{array}\right)\) (i) \(\mathrm{V}=\operatorname{span}(S)\) with the inner product \(\langle f, g\rangle=\int_{0}^{\pi} f(t) g(t) d t\), \(S=\\{\sin t, \cos t, 1, t\\}\), and \(h(t)=2 t+1\) (j) \(\mathrm{V}=\mathrm{C}^{4}, S=\\{(1, i, 2-i,-1),(2+3 i, 3 i, 1-i, 2 i)\), \((-1+7 i, 6+10 i, 11-4 i, 3+4 i)\\}\), and \(x=(-2+7 i, 6+9 i, 9-3 i, 4+4 i)\) (k) \(\mathrm{V}=\mathrm{C}^{4}, S=\\{(-4,3-2 i, i, 1-4 i)\), $$ (-1-5 i, 5-4 i,-3+5 i, 7-2 i),(-27-i,-7-6 i,-15+25 i,-7-6 i)\\}, $$ and \(x=(-13-7 i,-12+3 i,-39-11 i,-26+5 i)\) (1) \(\mathrm{V}=\mathrm{M}_{2 \times 2}(C), S=\left\\{\begin{array}{cc}1-i & -2-3 i \\ 2+2 i & 4+i\end{array}\right),\left(\begin{array}{cc}8 i & 4 \\\ -3-3 i & -4+4 i\end{array}\right)\), $$ \left.\left(\begin{array}{cc} -25-38 i & -2-13 i \\ 12-78 i & -7+24 i \end{array}\right)\right\\}, \text { and } A=\left(\begin{array}{rr} -2+8 i & -13+i \\ 10-10 i & 9-9 i \end{array}\right) $$ (m) \(\mathrm{V}=\mathrm{M}_{2 \times 2}(C), S=\left\\{\left(\begin{array}{cc}-1+i & -i \\ 2-i & 1+3 i\end{array}\right),\left(\begin{array}{cc}-1-7 i & -9-8 i \\ 1+10 i & -6-2 i\end{array}\right),\right.\), \(\left.\left(\begin{array}{cc}-11-132 i & -34-31 i \\ 7-126 i & -71-5 i\end{array}\right)\right\\}\), and \(A=\left(\begin{array}{cc}-7+5 i & 3+18 i \\ 9-6 i & -3+7 i\end{array}\right)\)

For each of the following inner product spaces \(\mathrm{V}\) (over \(F\) ) and linear transformations \(\mathrm{g}: \mathrm{V} \rightarrow F\), find a vector \(y\) such that \(\mathrm{g}(x)=\langle x, y\rangle\) for all \(x \in \mathrm{V}\). (a) \(\mathrm{V}=\mathrm{R}^{3}, \mathrm{~g}\left(a_{1}, a_{2}, a_{3}\right)=a_{1}-2 a_{2}+4 a_{3}\) (b) \(\mathrm{V}=\mathrm{C}^{2}, \mathrm{~g}\left(z_{1}, z_{2}\right)=z_{1}-2 z_{2}\) (c) \(\mathrm{V}=\mathrm{P}_{2}(R)\) with \(\langle f(x), h(x)\rangle=\int_{0}^{1} f(t) h(t) d t, \mathrm{~g}(f)=f(0)+f^{\prime}(1)\)

Let x = (2, 1 + i, i) and y = (2- i, 2. 1 + 2i) be vectors in C3 . Compute (x, y), llxll, IIYII. and llx + Yll· Then verify both the Cauchy- Schwarz inequality and the triangle inequality.

Prove that the composite of unitary [orthogonal] operators is unitary [orthogonal].

Let \(W\) be a finite-dimensional subspace of an inner product space \(V\). Show that if \(\mathrm{T}\) is the orthogonal projection of \(\mathrm{V}\) on \(\mathrm{W}\), then \(\mathrm{I}-\mathrm{T}\) is the orthogonal projection of \(\mathrm{V}\) on \(\mathrm{W}^{\perp}\).

In C2 , show that (x, y) = xAy• is an inner product, where $$ A=\left(\begin{array}{rr} 1 & i \\ -i & 2 \end{array}\right) $$ Compute \(\langle x, y\rangle\) for \(x=(1-i, 2+3 i)\) and \(y=(2+i, 3-2 i)\).

Which of the following pairs of matrices are unitarily equivalent? (a) \(\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)\) and \(\left(\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right)\) (b) \(\left(\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right)\) and \(\left(\begin{array}{ll}0 & \frac{1}{2} \\ \frac{1}{2} & 0\end{array}\right)\) (c) \(\left(\begin{array}{rrr}0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1\end{array}\right) \quad\) and \(\quad\left(\begin{array}{rrr}2 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 0\end{array}\right)\) (d) \(\left(\begin{array}{rrr}0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1\end{array}\right) \quad\) and \(\quad\left(\begin{array}{rrr}1 & 0 & 0 \\ 0 & i & 0 \\ 0 & 0 & -i\end{array}\right)\) (e) \(\left(\begin{array}{lll}1 & 1 & 0 \\ 0 & 2 & 2 \\ 0 & 0 & 3\end{array}\right)\) and \(\left(\begin{array}{lll}1 & 0 & 0 \\ 0 & 2 & 0 \\\ 0 & 0 & 3\end{array}\right)\)

Prove that if \(\mathrm{T}\) is a reflection on a 2-dimensional inner product space, then \(\mathrm{T}^{2}\) is the identity operator.

Prove that if \(\mathrm{T}\) is a unitary operator on a finite-dimensional inner product space \(\mathrm{V}\), then \(\mathrm{T}\) has a unitary square root; that is, there exists a unitary operator \(U\) such that \(T=U^{2}\). Visit goo.gl/jADTaS for a solution.