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

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) \(\left[\begin{array}{rrr}2 & -3 & -1 \\ 0 & 2 & 0 \\ 2 & 4 & 1\end{array}\right]\)

3

Find all possible linear combinations with value zero of the following sets of vectors. Then give the dimension of the spaces spanned by these sets of vectors.

  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 \(\mathcal{P}_{2} (the space of polynomials of degree at most 2)\).

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

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 \(\mathbb{R}^{3}\).

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

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 \(2 x-y+z=3\).

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

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

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