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?