List five vectors in Span $\{v1,v2\}$. For each vector, show the weights on $\mathbf v1$ and $\mathbf v2$ used to generate the vector and list the three entries of the vector. Do not make a sketch.
$$ {\mathbf v1} = \begin{bmatrix} 3 \\0 \\2 \\ \end{bmatrix}, {\mathbf v2} = \begin{bmatrix} -2 \\0 \\3 \\ \end{bmatrix} $$
I do not really understand what it means by weights I assume in might mean multiplication. I think the answer is:
$$v3 =\begin{bmatrix} 1 \\0 \\5 \\ \end{bmatrix} $$ Since I assume that you add them. How does one choose their weights?
Best Answer
Choose five pairs of scalars, $\{ a_1, a_2 \}$, and create the set of vectors consisting of $a_1 {\mathbf v1} + a_2 {\mathbf v2}$ using each such pair.