Proving...

• October 1st 2009, 10:03 AM
GreenDay14
Proving...

Prove that if (v1, ... , vn) is a basis of V, then so is(v1, v2-v1, ... , vn - vn-1).

I made the initial assumption that if V remained the same number that by subtracting v1, youd always end up with v1, and so on. But obviously I came to the conclusion that this would have to be wrong, because the set of (v1, .. , vn) could represent any numbers. So I have no idea how to tackle this one, any help would be greatly appreciated. Thanks.
• October 1st 2009, 10:11 AM
Jose27
If $a_i$ are such that $\sum_{i=1} ^{n} \ a_iv_i =0$ then $a_i =0$ for all $i$. Now suppose there are scalars $b_i$ such that $\sum_{i=1} ^{n} \ b_i(v_i-v_{i-1})=0$ where $v_0=0$ then we get $\sum_{i=1} ^{n} \ (b_i - b_{i+1})v_i=0$ where $b_{n+1}=0$ then $b_n=0$, and so $b_{n-1}=0$ ... $b_1=0$. Which means they're l.i. and therefore a basis.
• October 1st 2009, 11:00 AM
GreenDay14
Thank you very much Jose, but I am unsure as to whether I can approach the question like this. It would almost appear as if that is proving that they are a basis rather than specifically a basis for V. Would it be possible to get a second opinion on this?
• October 1st 2009, 11:06 AM
aman_cc
Any set of 'n' independent vectors which belong to a n-dimensional vector space,V, will be a basis of V.
(This is a std theorm and can be found in most of the texts on linear algebra)
• October 1st 2009, 11:25 AM
GreenDay14
Quote:

Originally Posted by aman_cc
Any set of 'n' independent vectors which belong to a n-dimensional vector space,V, will be a basis of V.
(This is a std theorm and can be found in most of the texts on linear algebra)

Yes I understand that it is a standard theorem, just as we can say X²=4 when x=2. But that is not what I am asking, I am asking how to PROVE this theorem. That is a completely different method. However I appreciate your input.
• October 1st 2009, 11:32 AM
Jose27
Quote:

Originally Posted by GreenDay14
Thank you very much Jose, but I am unsure as to whether I can approach the question like this. It would almost appear as if that is proving that they are a basis rather than specifically a basis for V. Would it be possible to get a second opinion on this?

What is your definition of basis?
• October 2nd 2009, 02:58 PM
HallsofIvy
Quote:

It would almost appear as if that is proving that they are a basis rather than specifically a basis for V.
You are given that the first n vectors form a basis for V so you know that V has dimension n. You are given another n vectors in V. To show that they are a basis for V you need only show that they span V or that they are independent.