# Thread: Need help with Extension Fields

1. ## Need help with Extension Fields

Prove that if V is a finite-dimensional vector space over a field F, then a subset {$\displaystyle \beta_{1}, \beta_{2}, ..., \beta_{n}$} of V is a basis for V over F if and only if every vector in V can be expressed uniquely as a linear combination of the $\displaystyle \beta_{i}$.

First things first. Could someone explain the concept of a linear combination? Wikipedia and my book aren't quite cutting it.

2. If you haven't grasped the idea of a linear combination, I highly doubt you understand what a basis is.

What is it that you don't understand? Where do you get stuck reading the wikipedia article?

3. What I've gathered from Wikipedia is that a linear combination is a set of scalars from a field multiplied by a set of vectors from a vector space. So in the literal sense of knowing the definition and knowing that a linear combination is the elements of a field multiplied by a the vectors in a vector space, I guess I understand it.

I don't understand what that does for us. And I have a rough thought of what a basis is, but probably not a well enough working idea to understand why that statement is true.