Can a Basis for a Subspace Always Include Specific Elements of a Larger Basis?

[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 ?

 

[slider142]

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

 

[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 ... .

 

[HallsofIvy]

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 ?

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.

 

[ehsan_thr]

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.
T 01:59 PM

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

 

[adriank]

Here's how I might approach the problem:

Start with an orthogonal basis of W, and pick any vector v in V that's not in W. Using your basis for W, can you use v to find a vector that's orthogonal to W?