Linear Algebra and Matrices, Subspaces, basis

[war485]

Homework Statement

I'm unclear about this statement being wrong or not:
if C is an x-dimensional subspace of Rⁿ, then a linearly independent set of x vectors in C is a basis for C

The Attempt at a Solution

I think that it must be a basis since it has independent vectors and it is in x dimensions, so there are x vectors in it. What I'm not so sure about is whether or not if it spans the subspace, but I think it does since all the vectors are independent, so there are x of them, so it should span in x dimensions.

I also want to thank HallsofIvy for lots of help with me with the questions 2 days ago.

 

[Dick]

Sure it's right. Isn't that the definition of dimension? The number of independent vectors required to span C?

 

[HallsofIvy]

If V is a vector space of dimension N, then a basis for V has three properties:
1) The vectors in the basis are independent.
2) The vectors span V.
3) There are N vectors in the basis.

If any two of those are true, the third is also.

 

[war485]

Thanks you two. :)