Exercise \(\PageIndex{1}\) In each case, determine which of axioms P1-P5 fail to hold.
a. \(V=\mathbb{R}^2,\left\langle\left(x_1, y_1\right),\left(x_2, y_2\right)\right\rangle=x_1 y_1 x_2 y_2\)
b.
\[
\begin{array}{l}
V=\mathbb{R}^3,\left\langle\left(x_1, x_2, x_3\right), \quad\left(y_1, y_2, y_3\right)\right\rangle=x_1 y_1- \\
x_2 y_2+x_3 y_3
\end{array}
\]
c. \(V=\mathbb{C},\langle z, w\rangle=z \bar{w}\), where \(\bar{w}\) is complex conjugation
d. \(V=\mathbf{P}_3,\langle p(x), q(x)\rangle=p(1) q(1)\)
e. \(V=\mathbf{M}_{22},\langle A, B\rangle=\operatorname{det}(A B)\)
f. \(V=\mathbf{F}[0,1],\langle f, g\rangle=f(1) g(0)+f(0) g(1)\)
Exercise \(\PageIndex{2}\) Let \(V\) be an inner product space. If \(U \subseteq V\) is a subspace, show that \(U\) is an inner product space using the same inner product.
Exercise \(\PageIndex{3}\) 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 \(f(x)=x^2\) \(\langle\mathbf{f}, \mathbf{g}\rangle \int_0^1 f(x) g(x) d x\)
b. \(\mathbf{v}=f\) in \(\mathbf{C}[-\pi, \pi]\) where \(f(x)=\cos x\) \(\langle\mathbf{f}, \mathbf{g}\rangle \int_{-\pi}^\pi f(x) g(x) d x\)
c. \(\mathbf{v}=\left[\begin{array}{l}1 \\ 3\end{array}\right]\) in \(\mathbb{R}^2\) where \(\langle\mathbf{v}, \quad \mathbf{w}\rangle=\) \(\mathbf{v}^T\left[\begin{array}{ll}1 & 1 \\ 1 & 2\end{array}\right] \mathbf{w}\)
d. \(\mathbf{v}=\left[\begin{array}{r}3 \\ -1\end{array}\right]\) in \(\mathbb{R}^2,\langle\mathbf{v}, \mathbf{w}\rangle=\mathbf{v}^T\left[\begin{array}{rr}1 & -1 \\ -1 & 2\end{array}\right] \mathbf{w}\)
Exercise \(\PageIndex{4}\) In each case, find the distance between \(\mathbf{u}\) and \(\mathbf{v}\).
a. \(\mathbf{u}=(3,-1,2,0), \mathbf{v}=(1,1,1,3) ;\langle\mathbf{u}, \mathbf{v}\rangle=\mathbf{u} \cdot \mathbf{v}\)
b. \(\mathbf{u}=(1,2,-1,2), \mathbf{v}=(2,1,-1,3) ;\langle\mathbf{u}, \mathbf{v}\rangle=\mathbf{u}\). \(\mathbf{v}\)
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 \(g(x)=\cos x ;\langle f, g\rangle=\int_{-\pi}^\pi f(x) g(x) d x\)
Exercise \(\PageIndex{5}\) Let \(a_1, a_2, \ldots, a_n\) be positive numbers. Given \(\mathbf{v}=\left(v_1, v_2, \ldots, v_n\right)\) and \(\mathbf{w}=\left(w_1\right.\), \(\left.w_2, \ldots, w_n\right)\), define \(\langle\mathbf{v}, \mathbf{w}\rangle=a_1 v_1 w_1+\ldots+a_n v_n w_n\). Show that this is an inner product on \(\mathbb{R}^n\).
Exercise \(\PageIndex{6}\) If \(\left\{\mathbf{b}_1, \ldots, \mathbf{b}_n\right\}\) is a basis of \(V\) and if \(\mathbf{v}=v_1 \mathbf{b}_1+\cdots+v_n \mathbf{b}_n\) and \(\mathbf{w}=w_1 \mathbf{b}_1+\cdots+w_n \mathbf{b}_n\) are vectors in \(V\), define
\[
\langle\mathbf{v}, \mathbf{w}\rangle=v_1 w_1+\cdots+v_n w_n .
\]
Show that this is an inner product on \(V\).
Exercise \(\PageIndex{7}\) If \(p=p(x)\) and \(q=q(x)\) are polynomials in \(\mathbf{P}_n\), define
\[
\langle p, q\rangle=p(0) q(0)+p(1) q(1)+\cdots+p(n) q(n)
\]
Show that this is an inner product on \(\mathbf{P}_n\). [Hint for P5: Theorem 6.5.4 or Appendix D.]
Exercise \(\PageIndex{8}\) Let \(\mathbf{D}_n\) denote the space of all functions from the set \(\{1,2,3, \ldots, n\}\) to \(\mathbb{R}\) with pointwise addition and scalar multiplication (see Exercise 35 Section 6.3). Show that \(\langle\),\(\rangle \) is an inner product on \(\mathbf{D}_n\) if \(\langle f, g\rangle=f(1) g(1)+f(2) g(2)+\) \(\ldots+f(n) g(n)\).
Exercise \(\PageIndex{9}\) Let re \((z)\) denote the real part of the complex number \(z\). Show that \(\langle\),\(\rangle\) is an inner product on \(\mathbb{C}\) if \(\langle z, w\rangle=r e(z \bar{w})\).
Exercise \(\PageIndex{10}\) If \(T: V \rightarrow V\) is an isomorphism of the inner product space \(V\), show that
\[
\langle\mathbf{v}, \mathbf{w}\rangle_1=\langle T(\mathbf{v}), T(\mathbf{w})\rangle
\]
defines a new inner product \(\langle,\rangle_1\) on \(V\).
Exercise (\PageIndex{11}\) Show that every inner product \(\langle\),\(\rangle\) on \(\mathbb{R}^n\) has the form \(\langle\mathbf{x}, \mathbf{y}\rangle=(U \mathbf{x}) \cdot(U \mathbf{y})\) for some upper triangular matrix \(U\) with positive diagonal entries. [Hint: Theorem 8.3.3.]
Exercise (\PageIndex{12}\) In each case, show that \(\langle\mathbf{v}, \mathbf{w}\rangle\) \(=\mathbf{v}^T A \mathbf{w}\) defines an inner product on \(\mathbb{R}^2\) and hence show that \(A\) is positive definite.
a. \(A=\left[\begin{array}{ll}2 & 1 \\ 1 & 1\end{array}\right]\)
b. \(A=\left[\begin{array}{rr}5 & -3 \\ -3 & 2\end{array}\right]\)
c. \(A=\left[\begin{array}{ll}3 & 2 \\ 2 & 3\end{array}\right]\)
d. \(A=\left[\begin{array}{ll}3 & 4 \\ 4 & 6\end{array}\right]\)
Exercise (\PageIndex{13}\) In each case, find a symmetric matrix \(A\) such that \(\langle\mathbf{v}, \mathbf{w}\rangle=\mathbf{v}^T A \mathbf{w}\).
a. \(\left.\left\langle\begin{array}{l}v_1 \\ v_2\end{array}\right],\left[\begin{array}{l}w_1 \\ w_2\end{array}\right]\right\rangle=v_1 w_1+2 v_1 w_2+\)
b. \(\left\langle\left[\begin{array}{l}v_1 \\ v_2\end{array}\right],\left[\begin{array}{l}w_1 \\ w_2\end{array}\right]\right\rangle=v_1 w_1-v_1 w_2-v_2 w_1+\)
c. \(\left\langle\left[\begin{array}{l}v_1 \\ v_2 \\ v_3\end{array}\right],\left[\begin{array}{l}w_1 \\ w_2 \\ w_3\end{array}\right]\right\rangle=2 v_1 w_1+v_2 w_2+\)
Exercise (\PageIndex{14}\) If \(A\) is symmetric and \(\mathbf{x}^T A \mathbf{x}=\) 0 for all columns \(\mathbf{x}\) in \(\mathbb{R}^n\), show that \(A=0\). [Hint: Consider \(\langle\mathbf{x}+\mathbf{y}, \mathbf{x}+\mathbf{y}\rangle\) where \(\langle\mathbf{x}, \mathbf{y}\rangle=\mathbf{x}^T A \mathbf{y}\).]
Exercise (\PageIndex{15}\) Show that the sum of two inner products on \(V\) is again an inner product.
Exercise (\PageIndex{16}\) Let \(\|\mathbf{u}\|=1,\|\mathbf{v}\|=2,\|\mathbf{w}\|=\sqrt{3}\), \(\langle\mathbf{u}, \mathbf{v}\rangle=-1,\langle\mathbf{u}, \mathbf{w}\rangle=0\) and \(\langle\mathbf{v}, \mathbf{w}\rangle=3\). Compute:
a. \(\langle\mathbf{v}+\mathbf{w}, 2 \mathbf{u}-\mathbf{v}\rangle\)
b. \(\langle\mathbf{u}-2 \mathbf{v}-\mathbf{w}, 3 \mathbf{w}-\mathbf{v}\rangle\)
Exercise (\PageIndex{17}\) Given the data in Exercise 16, show that \(\mathbf{u}+\mathbf{v}=\mathbf{w}\).
Exercise (\PageIndex{18}\) Show that no vectors exist such that \(\|\mathbf{u}\|=1,\|\mathbf{v}\|=2\), and \(\langle\mathbf{u}, \mathbf{v}\rangle=-3\).
Exercise (\PageIndex{19}\) Complete Example 10.1.2.
Exercise (\PageIndex{20}\) Prove Theorem 10.1.1.
Exercise (\PageIndex{21}\) Prove Theorem 10.1.6.
Exercise (\PageIndex{22}\) Let \(\mathbf{u}\) and \(\mathbf{v}\) be vectors in an inner product space \(V\).
a. Expand \(\langle 2 \mathbf{u}-7 \mathbf{v}, 3 \mathbf{u}+5 \mathbf{v}\rangle\).
b. Expand \(\langle 3 \mathbf{u}-4 \mathbf{v}, 5 \mathbf{u}+\mathbf{v}\rangle\).
c. Show that \(\|\mathbf{u}+\mathbf{v}\|^2=\|\mathbf{u}\|^2+2\langle\mathbf{u}, \mathbf{v}\rangle+\|\mathbf{v}\|^2\).
d. Show that \(\|\mathbf{u}-\mathbf{v}\|^2=\|\mathbf{u}\|^2-2\langle\mathbf{u}, \mathbf{v}\rangle+\|\mathbf{v}\|^2\).
Exercise (\PageIndex{23}\) Show that \(\|\mathbf{v}\|^2+\|\mathbf{w}\|^2=\) \(\frac{1}{2}\left\{\|\mathbf{v}+\mathbf{w}\|^2+\|\mathbf{v}-\mathbf{w}\|^2\right\}\) for any \(\mathbf{v}\) and \(\mathbf{w}\) in an inner product space.
Exercise (\PageIndex{24}\) Let \(\langle\),\(\rangle\) be an inner product on a vector space \(V\). Show that the corresponding distance function is translation invariant. That is, show that \(\mathrm{d}(\mathbf{v}, \mathbf{w})=\mathrm{d}(\mathbf{v}+\mathbf{u}, \mathbf{w}+\mathbf{u})\) for all \(\mathbf{v}, \mathbf{w}\), and \(\mathbf{u}\) in \(V\).
Exercise (\PageIndex{25}\) a. Show that \(\langle\mathbf{u}, \mathbf{v}\rangle=\frac{1}{4}\left[\|\mathbf{u}+\mathbf{v}\|^2=\|\mathbf{u}-\mathbf{v}\|^2\right]\) for all \(\mathbf{u}, \mathbf{v}\) in an inner product space \(V\).
b. If \(\langle\),\(\rangle and \langle,\rangle^{\prime}\) are two inner products on \(V\) that have equal associated norm functions, show that \(\langle\mathbf{u}, \mathbf{v}\rangle=\langle\mathbf{u}, \mathbf{v}\rangle^{\prime}\) holds for all \(\mathbf{u}\) and v.
Exercise (\PageIndex{26}\) Let \(\mathbf{v}\) denote a vector in an inner product space \(V\).
a. Show that \(W=\{\mathbf{w} \mid \mathbf{w}\) in \(V,\langle\mathbf{v}, \mathbf{w}\rangle=0\}\) is a subspace of \(V\).
b. If \(V=\mathbb{R}^3\) with the dot product, and if \(\mathbf{v}=(1\), \(-1,2)\), find a basis for \(W(W\) as in (a)).
Exercise (\PageIndex{27}\) Given vectors \(\mathbf{w}_1, \mathbf{w}_2, \ldots, \mathbf{w}_n\) and \(\mathbf{v}\), assume that \(\left\langle\mathbf{v}, \mathbf{w}_i\right\rangle=0\) for each \(i\). Show that \(\langle\mathbf{v}, \mathbf{w}\rangle=0\) for all \(\mathbf{w}\) in \(\operatorname{span}\left\{\mathbf{w}_1, \mathbf{w}_2, \ldots, \mathbf{w}_n\right\}\).
Exercise (\PageIndex{28}\) If \(V=\operatorname{span}\left\{\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n\right\}\) and \(\left\langle\mathbf{v}, \mathbf{v}_i\right\rangle=\left\langle\mathbf{w}, \mathbf{v}_i\right\rangle\) holds for each \(i\). Show that \(\mathbf{v}=\mathbf{w}\).
Exercise (\PageIndex{29}\) Use the Cauchy-Schwarz inequality in an inner product space to show that:
a. If \(\|\mathbf{u}\| \leq 1\), then \(\langle\mathbf{u}, \mathbf{v}\rangle^2 \leq\|\mathbf{v}\|^2\) for all \(\mathbf{v}\) in \(V\).
b. \((x \cos \theta+y \sin \theta)^2 \leq x^2+y^2\) for all real \(x, y\), and \(\theta\).
c. \(\left\|r_1 \mathbf{v}_1+\cdots+r_n \mathbf{v}_n\right\|^2 \leq\left[r_1\left\|\mathbf{v}_1\right\|+\cdots+\right.\) \(\left.r_n\left\|\mathbf{v}_n\right\|\right]^2\) for all vectors \(\mathbf{v}_i\), and all \(r_i>0\) in \(\mathbb{R}\).
Exercise (\PageIndex{30}\) If \(A\) is a \(2 \times n\) matrix, let \(\mathbf{u}\) and \(\mathbf{v}\) denote the rows of \(A\).
a. Show that \(A A^T=\left[\begin{array}{rr}\|\mathbf{u}\|^2 & \mathbf{u} \cdot \mathbf{v} \\ \mathbf{u} \cdot \mathbf{v} & \|\mathbf{v}\|^2\end{array}\right]\).
b. Show that \(\operatorname{det}\left(A A^T\right) \geq 0\).
Exercise (\PageIndex{31}\)
a. If \(\mathbf{v}\) and \(\mathbf{w}\) are nonzero vectors in an inner product space \(V\), show that \(-1 \leq \frac{\langle\mathbf{v}, \mathbf{w}\rangle}{\|\mathbf{v}\|\|\mathbf{w}\|} \leq 1\), and hence that a unique angle \(\theta\) exists such that \(\frac{\langle\mathbf{v}, \mathbf{w}\rangle}{\|\mathbf{v}\|\|\mathbf{w}\|}=\cos \theta\) and \(0 \leq \theta \leq \pi\). This angle \(\theta\) is called the angle between \(\mathbf{v}\) and \(\mathbf{w}\).
b. Find the angle between \(\mathbf{v}=(1,2,-1,1,3)\) and \(\mathbf{w}=(2,1,0,2,0)\) in \(\mathbb{R}^5\) with the dot product.
c. If \(\theta\) is the angle between \(\mathbf{v}\) and \(\mathbf{w}\), show that the law of cosines is valid:
\[
\|\mathbf{v}-\mathbf{w}\|=\|\mathbf{v}\|^2+\|\mathbf{w}\|^2-2\|\mathbf{v}\|\|\mathbf{w}\| \cos \theta .
\]
Exercise (\PageIndex{32}\) If \(V=\mathbb{R}^2\), define \(\|(x, y)\|=|x|+\) \(|y|\).
a. Show that \(\|\cdot\|\) satisfies the conditions in Theorem 10.1.5.
b. Show that \(\|\cdot\|\) does not arise from an inner product on \(\mathbb{R}^2\) given by a matrix \(A\). [Hint: If it did, use Theorem 10.1.2 to find numbers \(a\), \(b\), and \(c\) such that \(\|(x, y)\|^2=a x^2+b x y+c y^2\) for all \(x\) and \(y\).]
