# Thread: Linear Algebra - Direct Sum Problem.

1. ## Linear Algebra - Direct Sum Problem.

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!

2. Originally Posted by nlews
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!
What is it you're expected to prove?

To the moderators - through no fault of my own, my post was duplicated on this thread.

3. ## Response

Originally Posted by nlews
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!
What is it you're expected to prove?

4. 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, $B_U$, for U. Then there exist a basis $B_V$ for V which contains $B_U$. Then $B_V- B_U$ is also a basis for a subspace, $W$ of V and V is the direct sum of U and W. If u is any vector in U, then $B_U+ u$, by which I mean the set of vectors created by adding v to each vector in $B_U$, is a basis for a different subspace of V and V is the direct sum of U and this new subspace.

Examples in $R^3$:
Any non-trivial subspace of $R^3$ 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 $R^3$, W, consisting of any single line through the origin that is not contained in U is a subspace of $R^3$ and $R^3$ is the direct sum of U and W.

2) Let U be a line through the origin. Then the subset or $R^3$, W, consisting of any single plane through the origin that does not contain U is a supspace of $R^3$ and $R^3$ is the direct sum of U and W.

Example in $R^2$:
Any subspace of $R^2$ 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.