Activity2.2.4The equation \(A\xvec = \bvec\text{.}\)

  1. Consider the linear system

    \begin{equation*} \begin{alignedat}{4} 2x \amp {}+{} \amp y \amp {}-{} \amp 3z \amp {}={} \amp 4 \\ -x \amp {}+{} \amp 2y \amp {}+{} \amp z \amp {}={} \amp 3 \\ 3x \amp {}-{} \amp y \amp \amp \amp {}={} \amp -4 \\ \end{alignedat} \text{.} \end{equation*}

    Identify the matrix \(A\) and vector \(\bvec\) to express this system in the form \(A\xvec = \bvec\text{.}\)

  2. If \(A\) and \(\bvec\) are as below, write the linear system corresponding to the equation \(A\xvec=\bvec\text{.}\)

    \begin{equation*} A = \left[\begin{array}{rrr} 3 \amp -1 \amp 0 \\ -2 \amp 0 \amp 6 \end{array} \right], \bvec = \left[\begin{array}{r} -6 \\ 2 \end{array} \right] \end{equation*}

    and describe the solution space.

  3. Describe the solution space of the equation

    \begin{equation*} \left[ \begin{array}{rrrr} 1 \amp 2 \amp 0 \amp -1 \\ 2 \amp 4 \amp -3 \amp -2 \\ -1 \amp -2 \amp 6 \amp 1 \\ \end{array} \right] \xvec = \left[\begin{array}{r} -1 \\ 1 \\ 5 \end{array} \right] \text{.} \end{equation*}

  4. Suppose \(A\) is an \(m\times n\) matrix. What can you guarantee about the solution space of the equation \(A\xvec = \zerovec\text{?}\)

in-context