I'm having trouble understanding the difference between summing two subspaces and making ther union. My book says that the sum of two subspace is also a subspace, but I've found this example that shows that the union of a subspace is not always a subspace. So, what's the difference?
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1Do you know the definition of sum of two subspaces of a vector space? – Git Gud Nov 30 '14 at 23:58
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@GitGud $U + V = {u+v|u\in U, v \in V}$ – Guerlando OCs Nov 30 '14 at 23:59
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Can you prove that $U\cup V\subseteq U+V$ and can you see that the other inclusion doesn't hold a priori? – Git Gud Dec 01 '14 at 00:18
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$V= \mathbb{R}^2, W_1=\langle(0,1)\rangle, W_2=\langle(1,0)\rangle$
Then the union of $W_1$ and $W_2$ is the union of $x$ axis and $y$ axis.
But the sum of them, is all possible combinations of two elements in $W_1,W_2$, thus the sum is $V$, because we can write every element in $V$ as $(x,y)=x(1,0)+y(0,1)$.
HornyPigeon54
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