need help proving a statment with subspaces dimensions

• Jan 11th 2013, 10:47 AM
Stormey
need help proving a statment with subspaces dimensions
Hi.

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

• Jan 11th 2013, 11:32 AM
Deveno
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.
• Jan 11th 2013, 11:59 AM
vincisonfire
Re: need help proving a statment with subspaces dimensions
Deveno is right, your problem probably states $dim(U\cap W)\geq n-2$
• Jan 11th 2013, 12:33 PM
Stormey
Re: need help proving a statment with subspaces dimensions
sorry, let me rephrase that:

i need to prove that if $dim(V)=n$ and $dim(U)=dim(W)=n-1$, then $dim(U\cap W)=n-2$.
• Jan 11th 2013, 01:35 PM
Deveno
Re: need help proving a statment with subspaces dimensions
that's not true. see post #2.
• Jan 11th 2013, 02:03 PM
Stormey
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... (Doh)
• Jan 11th 2013, 04:24 PM
Deveno
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.
• Jan 12th 2013, 01:05 AM
Stormey
Re: need help proving a statment with subspaces dimensions
thanks again Deveno!
• Jan 14th 2013, 07:39 PM
LoidaWard
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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