Question 11.178

Calculate the dot product of \({\bf x}=(1,-1,0,2)\in {\mathbb R}^4\) and \({\bf y}=(1,2,3,4)\in{\mathbb R}^4\).

Question 11.179

In \({\mathbb R}^n\) show that

• (a) \(2 \|{\bf x} \|^2 + 2 \|{\bf y}\|^2 = \| {\bf x} + {\bf y}\|^2 + \|{\bf x} - {\bf y} \|^2\) (This is known as the parallelogram law.)
• (b) \(\|{\bf x} - {\bf y} \|\| {\bf x} + {\bf y}\|\le \|{\bf x} \|^2 + \|{\bf y} \|^2\)
• (c) \(4 \langle {\bf x}, {\bf y}\rangle = \|{\bf x} + {\bf y} \|^2 - \|{\bf x} - {\bf y}\|^2\) (This is called the polarization identity.)

Question 11.180

\({\bf x} = (2,0,-1), {\bf y} = (4,0,-2)\)

Question 11.181

\({\bf x} = (1,0,2,6), {\bf y} = (3,8,4,1)\)

Question 11.182

\({\bf x} = (1,-1,1, -1,1), {\bf y} =(3,0,0,0, 2)\)

Question 11.183

\({\bf x}=(1,0,0,1),{\bf y}=(-1,0,0,1)\)

Question 11.184

Let \(\textbf{v}, \textbf{w} \in \mathbb{R}^n\). If \( \| \textbf{v} \| = \| \textbf{w} \|\), show that \(\textbf{v} + \textbf{w}\) and \(\textbf{v} - \textbf{w}\) are orthogonal.

Question 11.185

Suppose \(T\) is a triangle formed by placing three points on a circle, two of which lie on the circle’s diameter. Use the previous problem to show \(T\) is a right triangle.

Question 11.186

Compute \(AB, \hbox{det } A, \hbox{det } B, \hbox{det } (A B)\), and \(\hbox{det } (A+B)\) for \[ A = \Bigg[\begin{array}{@{}c@{\quad}r@{\quad}c@{}} 1 & -1 & 0 \\ 0 & 3 & 2 \\ 3 & 1 & 1 \end{array} \Bigg] \qquad \hbox{and } \qquad B = \Bigg[\begin{array}{@{}r@{\quad}c@{\quad}r@{}} -2 & 0 & 2 \\ -1 & 1 & -1 \\ 1 & 4 & 3 \end{array} \Bigg]. \]

Question 11.187

Compute \(AB, \hbox{det } A, \hbox{det } B, \hbox{det } (AB)\), and \(\hbox{det } (A+B)\) for \[ A = \Bigg[\begin{array}{@{}c@{\quad}c@{\quad}r@{}} 3 & 0 & 1\\ 1 & 2 & -1 \\ 1 & 0 & 1 \end{array} \Bigg] \qquad \hbox{and } \qquad B = \Bigg[\begin{array}{@{}c@{\quad}c@{\quad}r@{}} 1 & 0 & -1 \\ 2 & 0 & 1 \\ 0 & 1 & 0 \end{array} \Bigg]. \]

Question 11.188

Determine which of the following matrices are invertible: \[ A=\left[ \begin{array}{ccc} 1 & 2 & 3 \\ 0 & 1 & 1 \\ 0 & 3 & 3 \\ \end{array} \right] \quad B=\left[ \begin{array}{ccc} 0 & 0 & 3 \\ -1 & 1 & 19 \\ 2 & 3 & \pi \\ \end{array} \right] \quad C=\left[ \begin{array}{cc} 1 & 1 \\ 1 & 1 \\ \end{array} \right] \]

Question 11.189

For matrix \(A\) in the previous problem, find a nonzero \(\textbf{x} \in \mathbb{R}^3\) such that \(A\textbf{x}=\textbf{0}\).

Question 11.190

Use induction on \(k\) to prove that if \({\bf x}_1, \ldots , {\bf x}_k \in {\mathbb R}^n\), then \[ \|{\bf x}_1 + \cdots + {\bf x}_k\| \le \|{\bf x}_1 \| + \cdots + \|{\bf x}_k\|. \]

Question 11.191

Using algebra, prove the identity of Lagrange: For real numbers \(x_1, \ldots , x_n\) and \(y_1, \ldots , y_n.\) \[ \bigg( \sum_{i=1}^n x_i y_i \bigg)^2\! = \bigg( \sum_{i=1}^n x_i^2 \bigg) \bigg( \sum_{i=1}^n y_i^2 \bigg) {-} \sum_{i < j} (x_i y_j - x_j y_i)^2. \]

Use this to give another proof of the Cauchy–Schwarz inequality in \({\mathbb R}^n\),

Question 11.192

Prove that if \(A\) is an \(n \times n \) matrix, then

• (a) \(\hbox{det } ( \lambda A) = \lambda^n \hbox{ det } A;\) and
• (b) if \(B\) is a matrix obtained from \(A\) by multiplying any row or column by a scalar \(\lambda\), then \(\hbox{det } B = \lambda \hbox{ det } A\).

Question 11.193

Is \(\hbox{det } (A+B) = \hbox{det } A + \hbox{det } B\)? Give a proof or counterexample.

Question 11.194

Does \((A+B) (A-B) = A^2 - B^2\)?

Question 11.195

Assuming the law \(\hbox{det } (AB) = ( \hbox{det } A)(\hbox{det } B)\), prove that det \((ABC) = ( \hbox{det } A) ( \hbox{det } B)(\hbox{det } C)\).

Question 11.196

(This exercise assumes a knowledge of integration of continuous functions of one variable.) Note that the proof of the Cauchy–Schwarz inequality (Theorem 4) depends only on the properties of the inner product listed in Theorem 1. Use this observation to establish the following inequality for continuous functions \(f,g \colon [0,1] \to {\mathbb R}\): \[ \bigg| \int_0^1 f (x) g (x)\ \ dx \bigg| \le \sqrt{ \int^1_0 [f (x) ]^2\ \ dx} \ \sqrt{\int_0^1[ g (x)]^2\ \ dx}. \]

Do this by

• (a) verifying that the space of continuous functions from [0, 1] to \({\mathbb R}\) forms a vector space; that is, we may think of functions \(f,g\) abstractly as “vectors” that can be added to each other and multiplied by scalars.
• (b) introducing the inner product of functions \[ f \,{\cdot}\, g = \int_0^1 f (x) g (x) \ dx \] and verifying that it satisfies conditions (i) to (iv) of Theorem 3.

Question 11.197

Define the transpose \(A^T\) of an \(n \times n\) matrix \(A\) as follows: the \(ij\)th element of \(A^T\) is \(a_{ji}\) where \(a_{ij}\) is the \(ij\)th entry of \(A\). Show that \(A^T\) is characterized by the following property: For all \({\bf x,y}\) in \({\mathbb R}^n\), \[ (A^T {\bf x}) \,{\cdot}\, {\bf y} = {\bf x} \,{\cdot}\, ( A {\bf y}). \]

Question 11.198

Verify that the inverse of \[ \Big[\begin{array} &a & b \\ c & d \end{array} \Big] \qquad \hbox{is} \qquad \frac{1}{ad- bc} \Big[\begin{array}{@{}r@{\quad}r@{}} d & -b \\ -c & a \end{array} \Big]. \]

Question 11.199

Use your answer in Exercise 21 to show that the solution of the system \begin{eqnarray*} && ax+ by = e \\ && cx + dy =f\\[-21.5pt] \end{eqnarray*} is \[ \Big[\begin{array}{c} x \\ y \end{array} \Big] = \frac{1}{ad-bc} \Big[\begin{array}{@{}r@{\quad}r@{}} d & -b \\ -c & a \end{array} \Big] \Big[\begin{array}{c} e \\ f \end{array} \Big]. \]

Question 11.200

Assuming the law \(\hbox{det } (AB) = ( \hbox{det } A) ( \hbox{det } B)\), verify that \((\hbox{det } A) (\hbox{det } A^{-1})=1\) and conclude that if \(A\) has an inverse, then \(\hbox{det } A \ne 0\).

Question 11.201

Find two \(2 \times 2\) matrices \(A\) and \(B\) such that \(AB=0\) but \(BA \neq 0\).