How do I find a way to write a a vector as a linear combination of other vectors 3 different ways?

Question

So I put this in my augmented matrix:

And then I find it's reduced echelon form:

But I'm not sure how I'm supposed to use this. Any ideas?

Answer 1

You want to find $a,b$ and $c$ such that

$ \begin{bmatrix} 2\\ 5 \end{bmatrix}=a\begin{bmatrix} 3\\ -1 \end{bmatrix}+b \begin{bmatrix} 1\\ 2\\ \end{bmatrix}+c \begin{bmatrix} 5\\ -3\\ \end{bmatrix}$

This can be turned into a system of equations:

$$3a+b+5c=2$$ $$-a+2b-3c=5$$

Then you used the augmented matrix and obtained that $$a+\frac{13}{7}c=\frac{-1}{7}$$ $$b-\frac{4}{7}c=\frac{17}{7}$$

This means that for any $c$ you choose, you can find $a$ and $b$ such that $ \begin{bmatrix} 2\\ 5 \end{bmatrix}$ is a linear combination of the other three matrices.

Indeed, for example choosing $c=1$, then we have $a=-2$ and $b=3$ so checking:

$a\begin{bmatrix} 3\\ -1 \end{bmatrix}+b \begin{bmatrix} 1\\ 2\\ \end{bmatrix}+c \begin{bmatrix} 5\\ -3\\ \end{bmatrix}=\begin{bmatrix} -6\\ 2 \end{bmatrix}+ \begin{bmatrix} 3\\ 6\\ \end{bmatrix}+ \begin{bmatrix} 5\\ -3\\ \end{bmatrix}=\begin{bmatrix} 2\\ 5 \end{bmatrix}$

as required.