[Math] Determine if b is in the span of the other given vectors.

Question

Determine if the vector $b$ is the span of the other given vector. If so, write $b$ as a linear combination of $a$
a=$\begin{bmatrix}
3\\ 5\end{bmatrix}$

b=$\begin{bmatrix}
9\\ -15\end{bmatrix}$

I am a bit lost on how to approach this question.

Answer 1

Is there any scalar $c$ such that $$c\;\begin{pmatrix} 3\\5\end{pmatrix} = \begin{pmatrix} 9\\ -15\end{pmatrix}\;\;?$$

That is, is it possible that both of the following equations hold: $$3c = 9\;\text{ AND }\;5c = -15$$ for the some fixed scalar $c$?

If not, then $b$ is not in the span of $a$.

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Note the difference in outcome if $\,b\,$ happened to be $\;\begin{pmatrix} 9\\15\end{pmatrix}$. Then, indeed, $b = 3a$.