# Linear Algebra - Direct Sum Problem.

• Sep 22nd 2010, 05:05 AM
nlews
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!
• Sep 22nd 2010, 08:20 AM
wonderboy1953
Quote:

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.
• Sep 22nd 2010, 08:20 AM
wonderboy1953
Response
Quote:

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?
• Sep 23rd 2010, 02:35 AM
HallsofIvy
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, \$\displaystyle B_U\$, for U. Then there exist a basis \$\displaystyle B_V\$ for V which contains \$\displaystyle B_U\$. Then \$\displaystyle B_V- B_U\$ is also a basis for a subspace, \$\displaystyle W\$ of V and V is the direct sum of U and W. If u is any vector in U, then \$\displaystyle B_U+ u\$, by which I mean the set of vectors created by adding v to each vector in \$\displaystyle 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 \$\displaystyle R^3\$:
Any non-trivial subspace of \$\displaystyle 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 \$\displaystyle R^3\$, W, consisting of any single line through the origin that is not contained in U is a subspace of \$\displaystyle R^3\$ and \$\displaystyle R^3\$ is the direct sum of U and W.

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

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