I'm doing some pre-term reading and I've come across direct sums which I'm having a bit of trouble getting my head around.
If V is a vector space and U is a proper subspace (non trivial), I'm told that there are infinitely many different subspaces W of V such that V=U(+)V
[ I've used (+) as the sign for direct sum as i'm not sure how to get it on my computer].
I don't really understand this, and would have no idea on how to prove it. The notes I'm reading hint at thinking about what happens with V is 2-dimensional. Any clues and help would be appreciated!
V is a vector space, U is a non-trivial subspace ("non-trivial" meaning here: there exist a non-zero vector in U, there exist a vector, w, in V that is not in U). Choose a basis, , for U. Then there exist a basis for V which contains . Then is also a basis for a subspace, of V and V is the direct sum of U and W. If u is any vector in U, then , by which I mean the set of vectors created by adding v to each vector in , is a basis for a different subspace of V and V is the direct sum of U and this new subspace.
Examples in :
Any non-trivial subspace of is either a line through the origin or a plane through the origin.
1) Let U be a plane containing the origin. Then the subset of , W, consisting of any single line through the origin that is not contained in U is a subspace of and is the direct sum of U and W.
2) Let U be a line through the origin. Then the subset or , W, consisting of any single plane through the origin that does not contain U is a supspace of and is the direct sum of U and W.
Example in :
Any subspace of is a line through the origin.
Let U be a line through the origin. Let W be any other line through the origin. Then V is the direct sum of U and W.