Question 1.145

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 1.146

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 1.147

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

Question 1.148

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

Question 1.149

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

Question 1.150

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

Question 1.151

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 1.152

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 1.153

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 1.154

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 1.155

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 1.156

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

Question 1.157

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 1.158

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 1.159

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 1.160

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

Question 1.161

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

Question 1.162

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 1.163

(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 1.164

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 1.165

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 1.166

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 1.167

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 1.168

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