:: Column space practise problems

  1. In each of the following,
    1. determine a basis for the column space of \(A\);
    2. express the other columns of \(A\) as linear combinations of the basis vectors from part i;
    3. if possible, express \(\mathbf{b}\) as a linear combination of the basis vectors from part i; and
    4. determine whether the system \(A \mathbf{x} = \mathbf{b}\) is consistent.
    1. \( A = \left[ \begin{array}{rr} 1 & 3 \\ 4 & -6 \end{array} \right] \), \( \mathbf{b} = \left[ \begin{array}{r} -2 \\ 10 \end{array} \right] \)
    2. \( A = \begin{bmatrix} 1 & 1 & 2 \\ 1 & 0 & 1 \\ 2 & 1 & 3 \end{bmatrix} \), \( \mathbf{b} = \left[ \begin{array}{r} -1 \\ 0 \\ 2 \end{array} \right] \)
    3. \( A = \left[ \begin{array}{rrr} 1 & -1 & 1 \\ 9 & 3 & 1 \\ 1 & 1 & 1 \end{array} \right] \), \( \mathbf{b} = \left[ \begin{array}{r} 5 \\ 1 \\ -1 \end{array} \right] \)
    4. \( A = \left[ \begin{array}{rrr} 1 & -1 & 1 \\ 1 & 1 & -1 \\ -1 & -1 & 1 \end{array} \right] \), \( \mathbf{b} = \begin{bmatrix} 2 \\ 0 \\ 0 \end{bmatrix} \)
    5. \( A = \begin{bmatrix} 1 & 2 & 0 & 1 \\ 0 & 1 & 2 & 1 \\ 1 & 2 & 1 & 3 \\ 0 & 1 & 2 & 2 \end{bmatrix} \), \( \mathbf{b} = \begin{bmatrix} 4 \\ 3 \\ 5 \\ 7 \end{bmatrix} \)
  2. In each of the following, a spanning set for a subspace of \(\mathbb{R}^4\) is given. Reduce the spanning set to a basis for the subspace; that is, determine a subset of the spanning set that is a basis for the subspace.
    1. \( \mathbf{v}_1 = (1,0,1,1) \), \( \mathbf{v}_2 = (-3,3,7,1) \), \( \mathbf{v}_3 = (-1,3,9,3) \), \( \mathbf{v}_4 = (-5,3,5,-1) \).
    2. \( \mathbf{v}_1 = (1,-2,0,3) \), \( \mathbf{v}_2 = (2,-4,0,6) \), \( \mathbf{v}_3 = (-1,1,2,0) \), \( \mathbf{v}_4 = (0,-1,2,3) \).
    3. \( \mathbf{v}_1 = (1,-1,5,2) \), \( \mathbf{v}_2 = (-2,3,1,0) \), \( \mathbf{v}_3 = (4,-5,9,4) \), \( \mathbf{v}_4 = (0,4,2,-3) \), \( \mathbf{v}_5 = (-7,18,2,-8) \).

Answers