Linear Independence and Linear Span
Solution set for Ax=0, Ax=B, and the dimension of null space/rank of a
matrix.

Def. linear independence over vector space (V, +, *)
Let v = {v1, ..., vk} be vectors in V. We say v is linearly independent
iff whenever a1v1 + a2v2 + ... akvk = 0, a1 = a2 = ... = ak = 0.

Where v_i in v in R^n, we can find linear independence by writing it as

[ v1 | v2 | ... | vk ][a1 a2 ... ak]^T = 0

Prove: basis of a vector space has the same dimension regardless of the
vectors.

Def basis: a linearly independent set of vectors which spans a vector
space.