MA 351 · Fall 2026 · Aaron N. K. Yip

Homework 3

Due: Thursday, Sept. 17, in class  ·  Penney, Linear Algebra: Ideas and Applications (4th ed.)

p. 67 Exercises
1.78

You are given a vector \(\mathbf{B}\) and vectors \(\mathbf{X}_i\). In each part, decide whether \(\mathbf{B}\) is in the span of the \(\mathbf{X}_i\) by attempting to solve the equation \(\mathbf{B} = x_1\mathbf{X}_1 + x_2\mathbf{X}_2 + x_3\mathbf{X}_3\).

1.79

In a vector space \(V\), a set of vectors spans \(V\) if every vector in \(V\) is a linear combination of the given vectors. Show that the vectors \(\mathbf{X} = [1,\,2]^t\) and \(\mathbf{Y} = [1,\,-2]^t\) span \(\mathbb{R}^2\). Specifically, show that the equation \([a,\,b]^t = x\mathbf{X} + y\mathbf{Y}\) is solvable regardless of \(a\) and \(b\).

1.80

Show that the vectors \(\mathbf{X}_1 = [3,\,1,\,5]^t\), \(\mathbf{X}_2 = [2,\,1,\,4]^t\), and \(\mathbf{X}_3 = [-1,\,2,\,3]^t\) do not span \(\mathbb{R}^3\) by finding a vector that cannot be expressed as a linear combination of them.

p. 86 Exercises
1.93

For each matrix \(A\) and each vector \(\mathbf{X}\), compute \(A\mathbf{X}\).

1.96

Find the nullspace for each of the matrices \(A\) in Exercise 1.93. Express each answer as a span.

1.108

The vectors \(\mathbf{X}_1 = [1,\,-1,\,1]^t\) and \(\mathbf{X}_2 = [1,\,0,\,1]^t\) span the same plane as \(\mathbf{Y}_1 = [4,\,-2,\,4]^t\) and \(\mathbf{Y}_2 = [0,\,-1,\,0]^t\). True or false? Explain.

1.109

The vectors \(\mathbf{X}_1 = [1,\,2,\,1,\,1]^t\), \(\mathbf{X}_2 = [1,\,1,\,1,\,1]^t\), \(\mathbf{X}_3 = [1,\,0,\,1,\,2]^t\) span the same subspace of \(\mathbb{R}^4\) as \(\mathbf{Y}_1 = [2,\,3,\,2,\,2]^t\), \(\mathbf{Y}_2 = [0,\,1,\,0,\,0]^t\), and \(\mathbf{Y}_3 = [1,\,1,\,1,\,1]^t\). True or false? Explain.

1.110

As CEO of an engineering firm, you have two groups of engineers working on solving the same linear system. Group I tells you that the solution is \[ [1,\,0,\,0]^t + s[-3,\,1,\,1]^t + t[-1,\,0,\,1]^t \] where \(s\) and \(t\) are arbitrary parameters. Group II tells you that the solution is \[ [1,\,-1,\,1]^t + s[-4,\,1,\,2]^t + t[-3,\,1,\,1]^t. \] Are the answers consistent? Explain.

1.121

Consider a system \(A\mathbf{X} = \mathbf{B}\), where \(\mathbf{B} \neq \mathbf{0}\). Let \(\mathbf{X}\) and \(\mathbf{Y}\) satisfy this system. Find all constants \(a\) and \(b\) such that \(a\mathbf{X} + b\mathbf{Y}\) also satisfies this system.

Hint/approach for 1.108 and 1.109

To show that \(\operatorname{Span}\{u_1, u_2, \ldots, u_k\} = \operatorname{Span}\{v_1, v_2, \ldots, v_l\}\) — note that in general \(k \neq l\) — you need to show the following two steps:

  1. For every given \(s_1, s_2, \ldots, s_k\), try to find \(t_1, t_2, \ldots, t_l\) such that \[ t_1 v_1 + t_2 v_2 + \cdots + t_l v_l = s_1 u_1 + s_2 u_2 + \cdots + s_k u_k, \] i.e., try to express the \(t_i\)'s in terms of the \(s_i\)'s; and
  2. For every given \(t_1, t_2, \ldots, t_l\), try to find \(s_1, s_2, \ldots, s_k\) such that \[ s_1 u_1 + s_2 u_2 + \cdots + s_k u_k = t_1 v_1 + t_2 v_2 + \cdots + t_l v_l, \] i.e., try to express the \(s_i\)'s in terms of the \(t_i\)'s.

Essentially, try to relate every \((s_1, s_2, \ldots, s_k)\) to \((t_1, t_2, \ldots, t_l)\) and vice versa.

Hint/approach for 1.110

This is similar to the above. Note that the first set of \(s, t\) might be different from the second set of \(s, t\). Essentially, you need to show that:

  1. For every given \(s_1, t_1\), try to find \(s_2, t_2\) such that \[ \begin{pmatrix}1\\-1\\1\end{pmatrix} + s_2\begin{pmatrix}-4\\1\\2\end{pmatrix} + t_2\begin{pmatrix}-3\\1\\1\end{pmatrix} = \begin{pmatrix}1\\0\\0\end{pmatrix} + s_1\begin{pmatrix}-3\\1\\1\end{pmatrix} + t_1\begin{pmatrix}-1\\0\\1\end{pmatrix}; \] and
  2. For every given \(s_2, t_2\), try to find \(s_1, t_1\) such that \[ \begin{pmatrix}1\\0\\0\end{pmatrix} + s_1\begin{pmatrix}-3\\1\\1\end{pmatrix} + t_1\begin{pmatrix}-1\\0\\1\end{pmatrix} = \begin{pmatrix}1\\-1\\1\end{pmatrix} + s_2\begin{pmatrix}-4\\1\\2\end{pmatrix} + t_2\begin{pmatrix}-3\\1\\1\end{pmatrix}. \]

Essentially, try to relate every \((s_1, t_1)\) to \((s_2, t_2)\) and vice versa.

Additional Problem

This problem lets you have a peek into the inner workings of the Google PageRank algorithm, in its most primitive form, uncorrupted by any human interference. It also shows that the homogeneous equation \(AX = 0\) is not unimportant.

The output of the PageRank algorithm is the ranking or relative importance of interconnected webpages. The (relative) importance of a webpage is given by a positive number and is determined by how many other webpages point to it. Let the (whole) internet consist of \(N\) webpages (nodes). Let \(x_i\) denote the (relative) importance of node \(i\). Then for \(i = 1, 2, \ldots, N\), \(x_i\) is given by: \[ x_i = \sum_j \frac{x_j}{D_j}, \] where the summation is over all the other webpages \(j\) that have a link to \(i\) and \(D_j\) is the total number of links (degree) that point out of \(j\). The division by \(D_j\) is such that if \(j\) points out to (million) many other webpages, then its contribution to the importance of any webpage it points to will be diminished. (We will assume that we do not consider any self-referencing of the webpages.)

Consider the following example of an internet with four webpages (nodes):

Four-node directed web graph. Nodes 1 (top-left), 2 (bottom-left), 3 (top-right), and 4 (bottom-right) are drawn as circles. Red arrows show directed edges: 1 to 3, 3 to 1, 1 to 4, 4 to 1, 2 to 3, 2 to 4, 1 to 2, and 4 to 3.
Figure: A directed graph representing an internet of four webpages. An arrow from node \(j\) to node \(i\) indicates that webpage \(j\) links to webpage \(i\). The actual links are: from webpage 1 to webapges 2, 3, and 4; from webpage 2 to webapges 3, and 4; from webpage 3 to webapge 1; from webpage 4 to webapges 1 and 3.

For example, Webpage 1 points out to Webpages 2, 3, and 4 so that \(D_1 = 3\), and only Webpages 3 and 4 point into Webpage 1. For Webpage 2, only Webpage 1 points to it and it points out only to Webpages 3 and 4 so that \(D_2 = 2\). Note also that \(D_3 = 1\) and \(D_4 = 2\). Now let \(x_1, x_2, x_3, x_4\) be the (relative) importance of the four webpages. Then we have \[ x_1 = x_3 + \frac{x_4}{2}; \qquad x_2 = \frac{x_1}{3}. \]

Write down the corresponding equations for \(x_3\) and \(x_4\). Put the overall \(4 \times 4\) system into a homogeneous form \(AX = 0\). Solve for \(x_1, x_2, x_3, x_4\). Note that you should be able to express your solution in terms of one free variable \(t\). Normalize your solution by choosing \(t\) such that \(x_1 + x_2 + x_3 + x_4 = 1\). Find the corresponding \(x_1, x_2, x_3, x_4\) using this \(t\). Then the ranking of the four webpages is given by the ranking of the \(x_i\)'s, i.e. the larger the \(x_i\), the more important is Webpage \(i\).

(You should be able to do this problem just with the description given above. But if you are curious, you can look at the article, The \$25,000,000,000 eigenvector: The linear algebra behind Google, by K. Bryan and T. Leise.)