Basis

Let be a subspace; a set of vectors forms a basis of if:

With a basis you can represent any vector in the subspace with a unique linear combination of the basis vectors.

Note

The dimension of , denoted as is the number of vectors in it's basis.

There are infinitely many choices for a basis in a non-zero subspace, this includes Orthogonal Basis and Orthonormal Basis. However all of these basis have the same number of vectors.

Example

Go to Finding Basis and Dimension to see a worked example of finding the basis of a span of vectors and it's dimension.

Quick Table summarizing the difference between a Basis, Orthogonal Basis, and an Orthonormal Basis.

Property Basis Orthogonal Basis Orthonormal Basis
Linearly independent x x x
Spans the subspace x x x
Vectors are perpendicular x x
Each vector has unit length x