Thread: linear algebra proof

Suppose that V is a vector space and that S = {u1,…,un} is one basis for V. Prove that the set T = {u1,…,un-1} is a linearly independent set but that it will not span V.

I think it is easy to prove that T is linearly independent, but how to conclude with it will not span V?

Thanks in advance!

Originally Posted by gavin1989

Suppose that V is a vector space and that S = {u1,…,un} is one basis for V. Prove that the set T = {u1,…,un-1} is a linearly independent set but that it will not span V.

I think it is easy to prove that T is linearly independent, but how to conclude with it will not span V?

Thanks in advance!

Your told that the vectors in T are independent, just prove that you cannot construct Un form the elements in T.

Bobak

V has a dimension of n. This means that you need n linearly independent vectors to form a basis of V.

However, in the new basis you only have n-1 vectors. Even if these vectors are linearly independent, they cannot span the space because you only have n-1 of them when you need n.

This was a theorem in my lecture notes, you might be able to find something similar in yours.