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An old exam I'm practicing with asks: $ W_1 = \left \{ \begin{bmatrix}a & a\\0 & b\end{bmatrix}: a,b \in R \right\} \ and \ W_2 = \left \{ \begin{bmatrix}c & d\\c & 0\end{bmatrix}: c,d \in R \right\}$

Find a basis of $ W_1 + W_2$ and of $W_1 \bigcap W_2$.

How do I add two 'spaces' together? Can I simply say that $W_1 + W_2$ is equal to $\left \{ \begin{bmatrix}a+c & a+d\\c & b\end{bmatrix}: a,b,c,d \in R \right\}$ It just seems too easy!

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    I understand what you are saying, but how do I actually show this? How do I demonstrate the space $W_1 + W_2$ represents? I remember something about unraveling them as vectors... is there such a thing or am I mixing that up with something else?2012-10-19

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Yeah, $U+V=\{u+v|u\in U, v\in V\}$.

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    So the attempt I made in my question then is wrong, since $a+c \ \ne a+d$. Right?2012-10-19