Proof dim(U+V)=dim U+dim V - dim(U∩V)

[b00tofuu]

let U and V be subspaces of Rn. Prove that dim(U+V)=dim U+dim V - dim(U∩V)

 

[HallsofIvy]

Okay, first, what is your definition of "dimension" of a vector space?

 

[b00tofuu]

its the number of vector in any basis for a subspace

 

[JSuarez]

Here's a hint: if U \cap V = \left\{0\right\}, then what must be the intersection of any base of U with any base of V?

 

[b00tofuu]

a point...?
but i still don't understand how to write the proof... T_T

 

[b00tofuu]

the zero point

 

[JSuarez]

Notice that the basis are SETS of vectors: if U \cap V = \left\{0\right\}, then the intersection of any base of U, with any base of V will be the empty set; also, dim\left(U\capV\right) = dim\left(\left\{0\right\}\right) = 0. Try this particular case first, then see if can generalize when U \cap V is not the null subspace.

 

[Tedjn]

One way to approach the problem is to ask yourself if you can find bases for the vector spaces U + V, U, V, and U ∩ V that are related somehow to each other.

 

[JSuarez]

You are complicating too much; it's simpler than that: consider a basis \left\{b_{i}\right\} for U \cap V; this basis can be extended to a basis \left\{u_{i}\right\} of U and \left\{v_{i}\right\} of V; now it's only a matter of counting the vectors.

 

[Tedjn]

I believe we're talking about the same thing. Extending a basis is what I meant by finding bases for U, V, and U ∩ V that are related to each other, then picking the right vectors from those bases to be a basis for U + V. Of course, I still might be missing something even simpler; it wouldn't be the first time :-)

 

[JSuarez]

Yes, but for proving that identity, not all all basis will do. Start with U \cap V.

 

[HallsofIvy]

Choose a basis for U. If any of those basis vectors are also in V, you can construct a basis for V including those vectors. If not, just choose any basis for V. Of course, the vectors in both bases, if any, form a basis for U\cap V. Now, just count!

How many vectors are there in the basis for U? How many vectors are there in the basis for V? How many are in both?