Vector spaces

[Filip84]

Hi,

can somebody check if this is correct:

Prove that in an n-dimensional vector space any set of n+1 vectors is linearly dependent.

Solution:

Vⁿ is vector space over field F.
Dimension of Vⁿ is n, which imply that basis B of Vⁿ is B={b₁, b₂, ..., b_n}. b₁, b₂, ..., bn are elements of Vⁿ , and they are linearly independent (by definition of Basis).


Hence we can write:

c₁b₁+c₂b₂+...c_nb_n=b_n+1 ,where c_i is an element of F.
Conclusion: Set {b₁, b₂, ..., b_n, b_n+1} is linearly dependent.

Thanks.

 

[daon2]

You are to prove any set of vectors is a linearly dependent set. You assumed something about your set of vectors, that a subset of them is a basis. With a little bit of tweaking you can make your proof work.

 

[Filip84]

Could you give me a hint?

 

[daon2]

Could you give me a hint?

I'll do it for 1 dimension to show you. Suppose B={b} is a basis for V, and consider a set of 2 (nonzero) vectors {v,w}.

Then v and w can be written as a linear combination of elements of B, i.e. v=s*b, w=t*b.

Consider the equation \(\displaystyle c_1v+c_2w=0\). Substituting we have \(\displaystyle c_1s\cdot b + c_2t\cdot b = (c_1s+c_2t)b=0\), i.e. \(\displaystyle c_1 = -c_2t/s\) gives a nontrivial solution to the equation (pick any value of c_2 you want), so {v,w} is linearly dependent.