Generating vectors in the span of two given vectors
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?
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$\begingroup$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.
$\endgroup$ 4 $\begingroup$I did not know the term weights in this context, but it is likely to mean the multiples $w_i$ of the spanning vectors $v_i$: $$ \DeclareMathOperator{span}{Span} \span\{v_1, v_2\} = \{ w_1 v_1 + w_2 v_2 \mid w_1, w_2 \in \mathbb{R} \} $$
Note: In other contexts, like linear optimization, there might be conditions imposed on those weight factors, e.g. $w_i \ge 0$ or $\sum_i w_i = 1$.
The resulting linear combinations would have special names, the first would result in conical combinations, all of them forming the conical hull, the second in affine combinations (affine hull), if both apply it results in convex combinations (convex hull).
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