# Thread: need help proving a statment with subspaces dimensions

1. ## need help proving a statment with subspaces dimensions

Hi.

Let $\displaystyle U,W$ be subspaces of $\displaystyle V$.
Suppose $\displaystyle dim(V)=n$, and $\displaystyle dim(U)=dim(W)=n-1$
I need to prove that $\displaystyle dim(U\cap W)=n-2$

2. ## Re: need help proving a statment with subspaces dimensions

this is not always TRUE.

if U = W, then dim(U) = dim(W) = n -1, but dim(U∩W) = n-1.

you can't prove things that are not true.

3. ## Re: need help proving a statment with subspaces dimensions

Deveno is right, your problem probably states $\displaystyle dim(U\cap W)\geq n-2$

4. ## Re: need help proving a statment with subspaces dimensions

sorry, let me rephrase that:

i need to prove that if $\displaystyle dim(V)=n$ and $\displaystyle dim(U)=dim(W)=n-1$, then $\displaystyle dim(U\cap W)=n-2$.

5. ## Re: need help proving a statment with subspaces dimensions

that's not true. see post #2.

6. ## Re: need help proving a statment with subspaces dimensions

Hi, Denovo.
i just noticed it says there that W and U are different.

sorry for the confusion.
the mid exams are next week, so i don't sleep much these days...

7. ## Re: need help proving a statment with subspaces dimensions

then it's a different story.

since W and U are distinct subspaces, say with bases B and B', respectively, then B' must contain a vector not in span(B). for if not, then U ⊆ W, in which case dim(U) = dim(W) forces U = W.

hence dim(U+W) = dim(span(B U B')) = n (glossed over a bit: if we call this vector uj, with B = {w1,...,wn-1}, B' = {u1,...,un-1}, we need to show:

C = BU{uj} is linearly independent. however, if:

c1w1+...+cn-1wn-1+cnuj = 0, we can distinguish 2 cases:

a) cn = 0. in this case, the linear independence of the wi forces c1 = ... = cn-1 = 0,

b) cn ≠ 0. in this case, uj = (-c1/cn)w1+...+(-cn-1/cn)wn-1, contradicting our choice of uj.

we also have n-1 = dim(span(B)) < dim(span(B U {uj}) ≤ dim(B U B') ≤ n).

therefore: dim(U∩W) = dim(U) + dim(W) - dim(U+W) = n-1 + n-1 - n = 2n - 2 - n = n - 2.

8. ## Re: need help proving a statment with subspaces dimensions

thanks again Deveno!

9. ## Re: need help proving a statment with subspaces dimensions

but for the converse one? So far, I only have a result that if K is a set consisting of all nonunits of R, then for any u, v in K, u+v won't be 1R. Does it help? Any suggestion?

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