# Basis of subspace

1. Feb 8, 2009

### ehsan_thr

suppose that {ei} 1<=i<=n is a basis for v and w is a subspace of v of dimension m<n ... . so Can we always find a basis for w which includes m elements of {ei} 1<=i<=n ???

does always w include m elements of {ei} 1<=i<=n ????

2. Feb 8, 2009

### slider142

What does it mean for a list of vectors to be the basis for a vector space V ?

3. Feb 8, 2009

### ehsan_thr

It means ,every vector of v is a linear combination of elements of the basis ... . ( The elements of v must be linear independent )
I have tried to prove my question , but i coudn't till now ... . my ultimate goal is to prove that for Every proper subspace W of a finite dimensional inner product space V there is a non zero vector x which is orothogonal to w ... .

4. Feb 8, 2009

### HallsofIvy

No. For example, let v be the set of linear polynomials and let {ei}= {x- 1, x+ 1}. Let w be the set of all multiples of x. Then w is a one dimensional subspace of v and any basis for w must be {ax} for some non-zero number a. None of those is in {ei}.

The other way is true: given any basis for subspace w we can extend it to a basis for v.

5. Feb 8, 2009

### ehsan_thr

Thank you , so much , i will find another solution for my problem ... .

6. Feb 8, 2009