Homework 8: Null Space and Column Space of a Matrix

Instructions. Assignments should be stapled and written clearly and legibly.

1. A matrix \(A\) and an echelon form of \(A\) are given: \begin{equation} A= \begin{bmatrix} \pe 1 & \pe 2 & -4 & \pe 3 & \pe 3\\ \pe 5 & \pe 10 & -9 & -7 & \pe 8\\ \pe 4 & \pe 8 & -9 & -2 & \pe 7\\ -2 & -4 & \pe 5 & \pe 0 & -6 \end{bmatrix} \sim \begin{bmatrix} \pe 1 & \pe 2 & -4 & \pe 3 & \pe 3\\ \pe 0 & \pe 0 & \pe 1 & -2 & \pe 0\\ \pe 0 & \pe 0 & \pe 0 & \pe 0 & -5\\ \pe 0 & \pe 0 & \pe 0 & \pe 0 & \pe 0 \end{bmatrix} \end{equation}
   1. Give any nonzero vector in \(\nul A\).
   2. Find a basis for \(\nul A\). What is \(\dim(\nul A)\)?
   3. Find a basis for \(\col A\). What is \(\dim(\col A)\)?
2. Without using a calculator or computer, find a nonzero vector in \(\nul A\), where \begin{equation}A = \begin{bmatrix} 51 & 51 & 58 & 2 & 7\\ 7 & 2049 & 9 & 1 & 2\\ 3 & 17 & 5 & 0 & 2\\ 9 & 2025 & 15 & 3 & 6\\ 3 & \sqrt{2} & 8 & 37 & 5\\ 7 & \pi & 23 & 19 & 16\\ 11 & 3.14 & 14 & 0 & 3 \end{bmatrix} \end{equation}
3. For each of the following vector spaces, find a matrix \(A\) such that the vector space is equal to \(\nul A\) . Then find a basis for the vector space.
   1. The line \(y=5x\) in \(\bR^2\).
   2. The plane \(x+2y+3z=0\) in \(\bR^3\).
4. Find a basis for \(\col \begin{bmatrix} 1&2\\ 0&3\\ 2&4 \end{bmatrix} \) without doing row reduction.
5. Find a basis for \(\col \begin{bmatrix} 1&3\\ 2&6\\ 3&9 \end{bmatrix} \) and then a basis for \(\nul \begin{bmatrix} 1&3\\ 2&6\\ 3&9 \end{bmatrix} \), both without doing row reduction.
6. Construct a \(2\times 3\) matrix \(C\) such that \(\nul C= \col \begin{bmatrix} 1 & 3 \\ 2 & 6 \\ 3 & 9 \end{bmatrix} \).
7. Construct a matrix \(A\) such that \(\col A= \nul \begin{bmatrix} 1 & 3 \\ 2 & 6 \\ 3 & 9 \end{bmatrix} \).
8. Let \( A = \begin{bmatrix} 1 & 0 & -3\\ 0 & 4 & -2\\ 2 & 6 & \pe 3 \end{bmatrix} \), and let \(\bv_1\), \(\bv_2\), and \(\bv_3\) be the three columns of \(A\).
   1. How many vectors are in Span\(\{\bv_1, \bv_2, \bv_3\}\)?
   2. How many vectors are in \(\col A\)?
   3. How many vectors are in \(\{\bv_1, \bv_2, \bv_3\}\)?
   4. Give two vectors in \(\col A\) which are not in \(\{\bv_1, \bv_2, \bv_3\}\)?
   5. Write the vector equation which is equivalent to the matrix equation \(A\mathbf{x} = \mathbf{0}\).
   6. Write the linear system of equations which is equivalent to the matrix equation \(A\mathbf{x} = \mathbf{0}\).
9. Let \(A\) be an \(m\times n\) matrix. Prove that \(\nul A=\{\bz\}\) if and only if the columns of \(A\) are linearly independent.