Basis (Vector Space)

linear-algebra

Definition

Basis (Vector Space)

Let be a vector space. A set is a basis of if is linearly independent and , where denotes the linear span of .

Every can be uniquely expressed as a linear combination of vectors from . The coefficients of this combination are the coordinates of with respect to .

Coordinate Projection

Translating coordinates between two different bases and of an -dimensional vector space , meaning translating coordinates into coordinates , can also be solved using matrix multiplication.

Formally, the coordinate translation is performed using a linear transformation , which is expressed as a matrix:

with columns .

Thus:

Note that is a coordinate projection.