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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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    It's because $W_1$ and $W_2$ are subspaces of $M_2(R)$ that you are allowed to do this.2012-10-18
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    Yes your answer is correct as @Peter wrote, and in fact if you look closer the space you just described is the full space of 2 by 2 matrices! (to proof this, just take an arbitrary matrix x,y,z,w and show that there are a,b,c,d such that it is of the form as you described.2012-10-18
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    by the way, this can be done easier, once you show that the intersection of the two spaces is zero, the sum of the spaces has dimension equal to the sum of the two spaces. They are both 2 dimensional (easy check), so their sum is 4 dimensional, which then must be the whole space. Hope i'm not reasoning in a way unfamiliar to you.2012-10-18
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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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