Prove: Existence of Basis for Finite Vector Space with Linearly Independent Set

[eyehategod]

Let S be a linearly independent set of vectors from the finite dimensional vector space V. Prove that there exists a basis for V containing S. Can anyone help me out? I can't figure out how to approach this.

 

[CompuChip]

What is a basis? What do you need to have one? What do you already have? Can you think of a way to construct the rest?

 

[HallsofIvy]

If the independent set already spans the space, you are done. If not then there exists a vector that cannot be written as a linear combination of the vectors in the independent set. Can you show that adding that new vector to the set still gives you an independent set of vectors? If that new set spans the vector space, you are done. If not ...