##### 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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