Homework 4

1

For the following matrices find the null space of \(I-A\) and find state vectors with nonnegative entries that sum to 1 in the null space, if any. Are these matrices stable (yes/no)?

  1. \(A=\left[\begin{array}{rrr}0.5 & 0 & 1 \\ 0.5 & 0.5 & 0 \\ 0 & 0.5 & 0\end{array}\right]\)

  2. \(A=\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]\)

Exercise 3.4.7

  1. \(\operatorname{span}\{(2,2,1)\},\left(\frac{2}{5}, \frac{2}{5}, \frac{1}{5}\right)\), yes

  2. \(\operatorname{span}\{(1,1)\},\left(\frac{1}{2}, \frac{1}{2}\right)\), no

2

Find bases for the row, column, and null space of each of the following matrices. (a) \([2,0,-1]\) (b) \(\left[\begin{array}{lllll}1 & 2 & 0 & 0 & 1 \\ 1 & 2 & 1 & 1 & 1 \\ 3 & 6 & 2 & 2 & 3\end{array}\right]\) (c) \(\left[\begin{array}{ccccc}1 & 2 & 0 & 4 & 0 \\ 1 & 3 & 5 & 2 & 1 \\ 2 & 3 & -5 & 10 & 0 \\ 2 & 4 & 0 & 8 & 1\end{array}\right]\) (d) \(\left[\begin{array}{rrr}2 & -3 & -1 \\ 0 & 2 & 0 \\ 2 & 4 & 1\end{array}\right]\)

Exercise 3.6.5

Bases for row, column, and null spaces:

  1. \(\{(2,0,-1)\},\{1\}\), \(\left\{\left(\frac{1}{2}, 0,1\right),(0,1,0)\right\}\)

  2. \((1,2,0,0,1),(0,0,1,1,0)\},\{(1,1,3),(0,1,2)\}\), \(\{(-2,1,0,0,0),(0,0,-1,1,0),(-1,0,0,0,1)\}\)

  3. \(\{(1,0,-10,8,0),(0,1,5,-2,0),(0,0,0,0,1)\}\), \(\{(1,1,2,2),(2,3,3,4),(0,1,0,1)\},\{(10,-5,1,0,0),(-8,2,0,1,0)\}\) (d) \(\left\{\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}\right\}\), \(\left\{\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}\right\},\{\}\)

3

Find all possible linear combinations with value zero of the following sets of vectors and the dimension of the space spanned by them.

  1. \((0,1,1),(2,0,1),(2,2,3),(0,2,2)\) in \(\mathbb{R}^{3}\).

  2. \(x, x^{2}+x, x^{2}-x\) in \(\mathcal{R}_{2}\).

  3. \((1,1,2,2),(0,2,0,2),(1,0,2,1),(2,1,4,4)\) in \(\mathbb{R}^{4}\).

Exercise 3.6.7

  1. \(c_{1} \mathbf{v}_{1}+c_{2} \mathbf{v}_{2}+c_{3} \mathbf{v}_{3}+c_{4} \mathbf{v}_{4}=\mathbf{0}\), where \(c_{1}=-2 c_{3}-2 c_{4}\), \(c_{2}=-c_{3}\), and \(c_{3}, c_{4}\) are free, dim \(\operatorname{span}\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}, \mathbf{v}_{4}\right\}=2\)

  2. \(c_{1} x+\) \(c_{2}\left(x^{2}+x\right)+c_{3}\left(x^{2}-x\right)=0\) where \(c_{1}=2 c_{3}, c_{2}=-c_{3}\), and \(c_{3}\) is free, dim span \(\left\{x, x^{2}+x, x^{2}-x\right\}=2\)

  3. \(c_{1} \mathbf{v}_{1}+c_{2} \mathbf{v}_{2}+c_{3} \mathbf{v}_{3}+c_{4} \mathbf{v}_{4}=\mathbf{0}\), where \(c_{1}=-c_{3}, c_{2}=\frac{1}{2} c_{3}, c_{4}=0\) and \(c_{3}\) is free, dim \(\operatorname{span}\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}, \mathbf{v}_{4}\right\}=3\)

4

Let \(A=\left[\begin{array}{lllll}0 & 1 & 0 & 1 & 2 \\ 1 & 0 & 2 & 1 & 2 \\ 2 & 2 & 4 & 4 & 8\end{array}\right]\). Use the column space algorithm on the matrix \([A I]\) to find a basis \(B\) of \(\mathcal{C}(A)\) and to expand it to a basis of \(\mathbb{R}^{3}\).

Exercise 3.6.13

\([A I]\) has RREF \(\left[\begin{array}{llllllcc}1 & 0 & 2 & 1 & 2 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 & 2 & 0 & -1 & \frac{1}{2} \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & -\frac{1}{2}\end{array}\right]\), so a basis of \(\mathcal{C}(A)\) is \(B=\)

\(\{(0,1,2),(1,0,2)\}\) and can be expanded to the basis \(\{(0,1,2),(1,0,2),(1,0,0)\}\) of \(\mathbb{R}^{3}\) according to the column space algorithm.

5

For the following orthogonal pairs \(\mathbf{u}, \mathbf{v}\) and matrix \(M=\left[\begin{array}{rrr}1 & 0 & -1 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]\), determine whether \(M \mathbf{u}\) and \(M \mathbf{v}\) are orthogonal.

  1. \((2,1,1),(1,0,-2)\)
  2. \((0,1,1),(1,-1,1)\)
  3. \((3,1,-2),(1,3,3)\)

4.2.7

  1. \((M \mathbf{u}) \cdot(M \mathbf{v})=1\), no
  2. \((M \mathbf{u}) \cdot(M \mathbf{v})=0\), yes
  3. \((M \mathbf{u})\). \((M \mathbf{v})=-13\), no

6

Find equations for the following planes in \(\mathbb{R}^{3}\).

  1. The plane containing the points \((1,1,2),(-1,3,2),(2,4,3)\).

  2. The plane containing the points \((-2,1,1)\) and \((0,1,2)\) and orthogonal to the plane \(2 x-y+z=3\).

Exercise 4.2.9

  1. \(x+y-4 z=-6\)

  2. \(x-2 z=-4\)

7

Find equations for the following hyperplanes in \(\mathbb{R}^{4}\).

  1. The plane parallel to the plane \(2 x_{1}+x_{2}-3 x_{3}+x_{4}=2\) and containing the point \((2,1,1,3)\).

  2. The plane through the origin and orthogonal to the vector \((1,0,2,1)\).

Exercise 4.2.10

  1. \(2 x_{1}+x_{2}-3 x_{3}+x_{4}=5\)

  2. \(x_{1}+2 x_{3}+x_{4}=0\)

8

For each pair \(A, \mathbf{b}\), solve the normal equations for the system \(A \mathbf{x}=\mathbf{b}\) and find the residual vector and its norm. Are there any genuine solutions to the system?

  1. \(\left[\begin{array}{ll}1 & 3 \\ 1 & 0\end{array}\right],\left[\begin{array}{l}1 \\ 3\end{array}\right]\)
  2. \(\left[\begin{array}{rr}2 & -2 \\ 1 & 1 \\ 3 & 1\end{array}\right],\left[\begin{array}{r}2 \\ -1 \\ 1\end{array}\right]\)
  3. \(\left[\begin{array}{rrr}0 & 2 & 2 \\ 1 & 1 & 0 \\ -1 & 1 & 2 \\ 1 & -2 & -3\end{array}\right],\left[\begin{array}{l}3 \\ 1 \\ 0 \\ 0\end{array}\right]\)

Exercise 4.2.11

  1. \(\mathbf{x}=\left(3,-\frac{2}{3}\right), \mathbf{b}-A \mathbf{x}=\mathbf{0},\|\mathbf{b}-A \mathbf{x}\|=0\), yes

  2. \(\mathbf{x}=\) \(\frac{1}{21}(9,-14), \mathbf{b}-A \mathbf{x}=\frac{1}{21}(-4,-16,8),\|\mathbf{b}-A \mathbf{x}\|=\frac{\sqrt{336}}{21}\), по

  3. \(\mathbf{x}=\) \(\left(x_{3}+\frac{12}{13},-x_{3}+\frac{23}{26}, x_{3}\right)\) where \(x_{3}\) is free, \(\mathbf{b}-A \mathbf{x}=\frac{1}{26}(32,-21,1,22),\|\mathbf{b}-A \mathbf{x}\|=\) \(\frac{\sqrt{1950}}{26}\), no

9

(Text retrieval) You are given the following term-by-document matrix, that is, a matrix whose \((i, j)\) th entry is the number of times term \(t_{i}\) occurs in document \(D_{j}\). Columns of this matrix are document vectors, as are queries. We measure the quality of a match between query and document by the cosine of the angle \(\theta\) between the two vectors, larger cosine being better. Which of the following nine documents \(D_{i}\) matches the query \((0,1,0,1,1)\) above the threshold value \(\cos \theta \geq 0.5\) ? Which is the best match to the query?

\(D_{1}\) \(D_{2}\) \(D_{3}\) \(D_{4}\) \(D_{5}\) \(D_{6}\) \(D_{7}\) \(D_{8}\) \(D_{9}\)
\(t_{1}\) 1 1 2 0 1 0 1 0 1
\(t_{2}\) 0 1 0 1 0 1 1 0 0
\(t_{3}\) 0 2 0 2 0 1 0 1 1
\(t_{4}\) 1 0 1 0 1 0 2 1 0
\(t_{5}\) 1 2 1 0 0 1 0 0 1

Exercise 4.2.14

Document 7 is the best match, with a cosine of \(\frac{3}{\sqrt{6} \sqrt{3}} \approx 0.70711\).