The question asks $\text{span}(A1,A2)$
$$A1 =\begin{bmatrix}1&2\\0&1\end{bmatrix}$$
$$A2 = \begin{bmatrix}0&1\\2&1\end{bmatrix}$$
I began by calculating $c_1[A1] + c_2[A2]$ then converting it into a matrix and row reducing. I found the restrictions where the stuff after the augment must = 0 then plugged those back into \begin{bmatrix}w&x\\y&z\end{bmatrix} and got the wrong answer. Could anyone please review my work and explain my mistake or the information I am missing? Thank you.
The solution I worked out (wrong):
Best Answer
Since you're using only matrix sum, instead of calculating with matrices, you can calculate with vectors (so $A_1 = (1,2,0,1)$, $A_2 = (0,1,2,1)$ and calculate the span as the span in $R^4$ (or whatever field you consider)- this is actually what you've at the end finished with (system of 4 linear equations with 4 variables), hence a problem in $R^4$ rather than in a matrix ring..
Looking at your solution, it seems to me that you have a typo at the very beggining - the $y$ entry is equal to $2 c_2 + 0 c_1$.