Problem 3
Exercise 10.1 .3 In each case, find a scalar multiple of \(\mathbf{v}\) that is a unit vector. a. \(\mathbf{v}=f\) in \(\mathbf{C}[0,1]\) where \(f(x)=x^{2}\) $$ \langle f, g\rangle \int_{0}^{1} f(x) g(x) d x $$ b. \(\mathbf{v}=f\) in \(\mathbf{C}[-\pi, \pi]\) where \(f(x)=\cos x\) \(\langle f, g\rangle \int_{-\pi}^{\pi} f(x) g(x) d x\) c. \(\mathbf{v}=\left[\begin{array}{l}1 \\ 3\end{array}\right]\) in \(\mathbb{R}^{2}\) where \(\langle\mathbf{v}, \mathbf{w}\rangle=\mathbf{v}^{T}\left[\begin{array}{ll}1 & 1 \\ 1 & 2\end{array}\right] \mathbf{w}\) d. \(\mathbf{v}=\left[\begin{array}{r}3 \\ -1\end{array}\right]\) in \(\mathbb{R}^{2},\langle\mathbf{v}, \mathbf{w}\rangle=\mathbf{v}^{T}\left[\begin{array}{rr}1 & -1 \\ -1 & 2\end{array}\right] \mathbf{w}\)
Problem 3
\(\begin{array}{llll}\text { Exercise } & \mathbf{1 0 . 2 . 3} & \text { Let } & \mathbf{M}_{22} \text { have the inner product }\end{array}\) \(\langle X, Y\rangle=\operatorname{tr}\left(X Y^{T}\right) .\) In each case, use the Gram- Schmidt algorithm to transform \(B\) into an orthogonal basis. a. \(B=\left\\{\left[\begin{array}{ll}1 & 1 \\ 0 & 0\end{array}\right],\left[\begin{array}{ll}1 & 0 \\ 1 & 0\end{array}\right],\left[\begin{array}{ll}0 & 1 \\ 0 & 1\end{array}\right],\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\right\\}\) b. \(B=\left\\{\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right],\left[\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right],\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right],\left[\begin{array}{ll}1 & 0 \\ 0 & 0\end{array}\right]\right\\}\)
Problem 4
Exercise 10.1 .4 In each case, find the distance between \(\mathbf{u}\) and \(\mathbf{v}\). $$ \begin{array}{l} \text { a. } \mathbf{u}=(3,-1,2,0), \mathbf{v}=(1,1,1,3) ;\langle\mathbf{u}, \mathbf{v}\rangle=\mathbf{u} \cdot \mathbf{v} \\ \text { b. } \mathbf{u}=(1,2,-1,2), \mathbf{v}=(2,1,-1,3) ;\langle\mathbf{u}, \mathbf{v}\rangle=\mathbf{u} \cdot \mathbf{v} \end{array} $$ c. \(\mathbf{u}=f, \mathbf{v}=g\) in \(\mathbf{C}[0,1]\) where \(f(x)=x^{2}\) and \(g(x)=1-x ;\langle f, g\rangle=\int_{0}^{1} f(x) g(x) d x\) d. \(\mathbf{u}=f, \mathbf{v}=g\) in \(\mathbf{C}[-\pi, \pi]\) where \(f(x)=1\) and \(g(x)=\cos x ;\langle f, g\rangle=\int_{-\pi}^{\pi} f(x) g(x) d x\)
Problem 4
\(V\) denotes a finite dimensional inner product space. Let \(T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) be an isometry. A vector \(\mathbf{x}\) in \(\mathbb{R}^{2}\) is said to be fixed by \(T\) if \(T(\mathbf{x})=\mathbf{x}\). Let \(E_{1}\) denote the set of all vectors in \(\mathbb{R}^{2}\) fixed by \(T\). Show that: a. \(E_{1}\) is a subspace of \(\mathbb{R}^{2}\). b. \(E_{1}=\mathbb{R}^{2}\) if and only if \(T=1\) is the identity map. c. \(\operatorname{dim} E_{1}=1\) if and only if \(T\) is a reflection (about the line \(E_{1}\) ). d. \(E_{1}=\\{0\\}\) if and only if \(T\) is a rotation \((T \neq 1)\).
Problem 4
Let \(V\) be an \(n\) -dimensional inner product space, and let \(T\) and \(S\) denote symmetric linear operators on \(V\). Show that: a. The identity operator is symmetric. b. \(r T\) is symmetric for all \(r\) in \(\mathbb{R}\). c. \(S+T\) is symmetric. d. If \(T\) is invertible, then \(T^{-1}\) is symmetric. e. If \(S T=T S,\) then \(S T\) is symmetric.
Problem 4
Show that \(\\{1, \cos x, \cos (2 x), \cos (3 x), \ldots\\}\) is an orthogonal set in \(\mathbf{C}[0, \pi]\) with respect to the inner product \(\langle f, g\rangle=\int_{0}^{\pi} f(x) g(x) d x\).
Problem 4
In each case, use the Gram-Schmidt process to convert the basis \(B=\left\\{1, x, x^{2}\right\\}\) into an orthogonal basis of \(\mathbf{P}_{2}\). $$ \begin{array}{l} \text { a. }\langle p, q\rangle=p(0) q(0)+p(1) q(1)+p(2) q(2) \\ \text { b. }\langle p, q\rangle=\int_{0}^{2} p(x) q(x) d x \end{array} $$
Problem 5
Show that \(\left\\{1, x-\frac{1}{2}, x^{2}-x+\frac{1}{6}\right\\},\) is an orthogonal basis of \(\mathbf{P}_{2}\) with the inner product $$ \langle p, q\rangle=\int_{0}^{1} p(x) q(x) d x $$ and find the corresponding orthonormal basis.
Problem 5
In each case, show that \(T\) is symmetric and find an orthonormal basis of eigenvectors of \(T\). a. \(T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}\) \(T(a, b, c)=(2 a+2 c, 3 b, 2 a+5 c) ;\) use the dot product b. \(T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}\) \(T(a, b, c)=(7 a-b,-a+7 b, 2 c) ;\) use the dot product c. \(T: \mathbf{P}_{2} \rightarrow \mathbf{P}_{2}\) \(\quad T\left(a+b x+c x^{2}\right)=3 b+(3 a+4 c) x+4 b x^{2}\) inner product \(\left\langle a+b x+c x^{2}, a^{\prime}+b^{\prime} x+c^{\prime} x^{2}\right\rangle=a a^{\prime}+b b^{\prime}+c c^{\prime}\) d. \(T: \mathbf{P}_{2} \rightarrow \mathbf{P}_{2}\) \(\quad T\left(a+b x+c x^{2}\right)=(c-a)+3 b x+(a-c) x^{2} ;\) inner product as in part (c)
Problem 5
Let \(a_{1}, a_{2}, \ldots, a_{n}\) be positive numbers. Given \(\mathbf{v}=\left(v_{1}, v_{2}, \ldots, v_{n}\right)\) and \(\mathbf{w}=\left(w_{1}, w_{2}, \ldots, w_{n}\right)\) define \(\langle\mathbf{v}, \mathbf{w}\rangle=a_{1} v_{1} w_{1}+\cdots+a_{n} v_{n} w_{n} .\) Show that this is an inner product on \(\mathbb{R}^{n}\).
