How to prove that $a \begin{pmatrix} 3 \\ -1 \end{pmatrix} + b \begin{pmatrix} 2 \\ 7 \end{pmatrix}$ is true for any real number $a$, and $b$?

Question

Show that any two-dimensional vector can be expressed in the form $a \begin{pmatrix} 3 \\ -1 \end{pmatrix} + b \begin{pmatrix} 2 \\ 7 \end{pmatrix}$, where $a$ and $b$ are real numbers.

I was able to simplify this down to: $\binom{3a+2b}{-a+7b}=\binom{x}{y}\\$, where $\binom{x}{y} = a\binom{3}{-1} + b \binom{2}{7}$, but I am not sure what to do from here.

Answer 1

You are almost there.

You have $\binom{3a+2b}{-a+7b}=\binom{x}{y}\\ $.

This is the set of equations $3a+2b = x$ and $-a+7b = y$.

Just solve these for $a$ and $b$ in terms of $x$ and $y$.

Answer 2

Note that for all $x,y \in \mathbb{R}$, if we solve $3a+2b=x$ and $-a+7b=y$, we get that $$a=\frac{7x}{23}-\frac{2y}{23}$$$$b=\frac{x}{23}+\frac{3y}{23}$$ We have then that $$a \begin{pmatrix} 3 \\ -1 \end{pmatrix} + b \begin{pmatrix} 2 \\ 7 \end{pmatrix}=\begin{pmatrix} 3a+2b \\ -a+7b \end{pmatrix}=\begin{pmatrix} x \\ y \end{pmatrix}$$ Which follows from calculations.