Activity 2.1.3

Given the vectors $$\vvec_1 = \left[\begin{array}{r} 4 \\ 0 \\ 2 \\ 1 \end{array} \right], \vvec_2 = \left[\begin{array}{r} 1 \\ -3 \\ 3 \\ 1 \end{array} \right], \vvec_3 = \left[\begin{array}{r} -2 \\ 1 \\ 1 \\ 0 \end{array} \right], \bvec = \left[\begin{array}{r} 0 \\ 1 \\ 2 \\ -2 \end{array} \right]$$, can $\bvec$ be expressed as a linear combination of $\vvec_1$, $\vvec_2$, and $\vvec_3$? Rephrase this question by writing a linear system for the weights $c_1$, $c_2$, and $c_3$ and use the Sage cell below to answer this question.

Consider the following linear system. $$\begin{alignedat}{4} 3x_1 & {}+{} & 2x_2 & {}-{} x_3 & {}={} & 4 \\ x_1 & & & {}+{} 2x_3 & {}={} & 0 \\ -x_1 & {}-{} & x_2 & {}+{} 3x_3 & {}={} & 1 \\ \end{alignedat}$$ Identify vectors $\vvec_1$, $\vvec_2$, $\vvec_3$, and $\bvec$ such that the question "Is this linear system consistent?" is equivalent to the question "Can $\bvec$ be expressed as a linear combination of $\vvec_1$, $\vvec_2$, and $\vvec_3$?"

Consider the vectors $$\vvec_1 = \left[\begin{array}{r} 0 \\ -2 \\ 1 \\ \end{array} \right], \vvec_2 = \left[\begin{array}{r} 1 \\ 1 \\ -1 \\ \end{array} \right], \vvec_3 = \left[\begin{array}{r} 2 \\ 0 \\ -1 \\ \end{array} \right], \bvec = \left[\begin{array}{r} -1 \\ 3 \\ -1 \\ \end{array} \right]$$. Can $\bvec$ be expressed as a linear combination of $\vvec_1$, $\vvec_2$, and $\vvec_3$? If so, can $\bvec$ be written as a linear combination of these vectors in more than one way?

Considering the vectors $\vvec_1$, $\vvec_2$, 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 $\left[\begin{array}{rrr} \vvec_1 & \vvec_2 & \vvec_3 \end{array} \right]$ help answer this question.

Now consider the vectors $$\vvec_1 = \left[\begin{array}{r} 0 \\ -2 \\ 1 \\ \end{array} \right], \vvec_2 = \left[\begin{array}{r} 1 \\ 1 \\ -1 \\ \end{array} \right], \vvec_3 = \left[\begin{array}{r} 1 \\ -1 \\ -2 \\ \end{array} \right], \bvec = \left[\begin{array}{r} 0 \\ 8 \\ -4 \\ \end{array} \right]$$. Can $\bvec$ be expressed as a linear combination of $\vvec_1$, $\vvec_2$, and $\vvec_3$? If so, can $\bvec$ be written as a linear combination of these vectors in more than one way?

Considering the vectors $\vvec_1$, $\vvec_2$, 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 $\left[\begin{array}{rrr} \vvec_1 & \vvec_2 & \vvec_3 \end{array} \right]$ help answer this question.