Q:

Creating a Basis with constraints

I'm trying to find a basis for $R^4$ space in the form $(a, b, c, d)$ where $d=a+b$ and $c=a-b$.
I can answer $(a)$ and $(c)$ but part $(b)$ is giving me a bit of trouble https://i.imgur.com/pRGG7A1.png
I'm not really sure where to start. The answer sheet lists $(a + b + (a+b) + (a-b))$ = $a(1, 0, 1, 1) + b(0, 1, 1, -1)$.
From this I can gather that they've substituted the variables for $c=(a+b)$ and $d=(a-b)$ into the equation but I can't for the life of me see where the $a(1, 0, 1, 1) + b(0, 1, 1, -1)$ comes from.
Thanks in advance for your help!

A:

Every vector in that subspace can be written as
$$\begin{bmatrix}a\\b\\c\\d\end{bmatrix}
= \begin{bmatrix}a\\b\\a-b\\a+b\end{bmatrix}
= a \begin{bmatrix}1\\0\\1\\1\end{bmatrix}
+ b \begin{bmatrix}0\\1\\-1\\1\end{bmatrix}.$$

