Activity2.1.3Linear combinations and linear systems

  1. Given the vectors

    \begin{equation*} \vvec_1 = \left[\begin{array}{r} 4 \\ 0 \\ 2 \\ 1 \end{array} \right], \vvec_2 = \left[\begin{array}{r} 1 \\ -3 \\ 3 \\ 1 \end{array} \right], \vvec_3 = \left[\begin{array}{r} -2 \\ 1 \\ 1 \\ 0 \end{array} \right], \bvec = \left[\begin{array}{r} 0 \\ 1 \\ 2 \\ -2 \end{array} \right] \text{,} \end{equation*}

    we ask if \(\bvec\) can be expressed as a linear combination of \(\vvec_1\text{,}\) \(\vvec_2\text{,}\) and \(\vvec_3\text{.}\) Rephrase this question by writing a linear system for the weights \(c_1\text{,}\) \(c_2\text{,}\) and \(c_3\) and use the Sage cell below to answer this question.

  2. Consider the following linear system.

    \begin{equation*} \begin{alignedat}{4} 3x_1 \amp {}+{} \amp 2x_2 \amp {}-{} x_3 \amp {}={} \amp 4 \\ x_1 \amp \amp \amp {}+{} 2x_3 \amp {}={} \amp 0 \\ -x_1 \amp {}-{} \amp x_2 \amp {}+{} 3x_3 \amp {}={} \amp 1 \\ \end{alignedat} \end{equation*}

    Identify vectors \(\vvec_1\text{,}\) \(\vvec_2\text{,}\) \(\vvec_3\text{,}\) and \(\bvec\) and rephrase the question "Is this linear system consistent?" by asking "Can \(\bvec\) be expressed as a linear combination of \(\vvec_1\text{,}\) \(\vvec_2\text{,}\) and \(\vvec_3\text{?}\)"

  3. Consider the vectors

    \begin{equation*} \vvec_1 = \left[\begin{array}{r} 0 \\ -2 \\ 1 \\ \end{array} \right], \vvec_2 = \left[\begin{array}{r} 1 \\ 1 \\ -1 \\ \end{array} \right], \vvec_3 = \left[\begin{array}{r} 2 \\ 0 \\ -1 \\ \end{array} \right], \bvec = \left[\begin{array}{r} -1 \\ 3 \\ -1 \\ \end{array} \right] \text{.} \end{equation*}

    Can \(\bvec\) be expressed as a linear combination of \(\vvec_1\text{,}\) \(\vvec_2\text{,}\) and \(\vvec_3\text{?}\) If so, can \(\bvec\) be written as a linear combination of these vectors in more than one way?

  4. Considering the vectors \(\vvec_1\text{,}\) \(\vvec_2\text{,}\) and \(\vvec_3\) from the previous part, can we write every three-dimensional vector \(\bvec\) as a linear combination of these vectors? Explain how the pivot positions of the matrix \(\left[\begin{array}{rrr} \vvec_1 \amp \vvec_2 \amp \vvec_3 \end{array} \right]\) help answer this question.

  5. Now consider the vectors

    \begin{equation*} \vvec_1 = \left[\begin{array}{r} 0 \\ -2 \\ 1 \\ \end{array} \right], \vvec_2 = \left[\begin{array}{r} 1 \\ 1 \\ -1 \\ \end{array} \right], \vvec_3 = \left[\begin{array}{r} 1 \\ -1 \\ -2 \\ \end{array} \right], \bvec = \left[\begin{array}{r} 0 \\ 8 \\ -4 \\ \end{array} \right] \text{.} \end{equation*}

    Can \(\bvec\) be expressed as a linear combination of \(\vvec_1\text{,}\) \(\vvec_2\text{,}\) and \(\vvec_3\text{?}\) If so, can \(\bvec\) be written as a linear combination of these vectors in more than one way?

  6. Considering the vectors \(\vvec_1\text{,}\) \(\vvec_2\text{,}\) and \(\vvec_3\) from the previous part, can we write every three-dimensional vector \(\bvec\) as a linear combination of these vectors? Explain how the pivot positions of the matrix \(\left[\begin{array}{rrr} \vvec_1 \amp \vvec_2 \amp \vvec_3 \end{array} \right]\) help answer this question.

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