Linear Algebra/Basis Vectors

Definitions

A basis of a vector space V is a set of vectors which have the following properties:

• They are linearly independent.
• Their linear combinations build up every vector of V.

A vector space is of dimension d if there exists d linearly independent vectors and that any d+1 vectors are linearly dependent.

Remarkable theorems

Theorem

In a vector space of dimension d, any d linearly independent vectors form a basis for that vector space.

Proof

Let there be d vectors. Let x be another vector. Then those d vectors and x are linearly dependent, so x is linearly dependent on those d vectors. Hence, those d vectors form a basis.

Theorem

If a vector space has d vectors for a basis, then it is of dimension d.

Theorem (completion)

If you have m linearly independent vectors in a vector space of dimension n (with m<=n), then you can choose n-m vectors which form a basis of the vector space along with the starting m vectors.

Proof

Those m vectors do not form a basis since it is not equal to n, so there exists a vector in the vector space linearly independent of them. Continuing choosing vectors independent of the previous ones in this fashion until one has n vectors.