32. Coordinate Vector

Given a basis of a vector space , every has a unique representation as a linear combination of the basis vectors:

The coordinate vector of with respect to collects those coefficients:

The bracket-subscript notation emphasizes that the coordinates depend on the basis.

32.1. The standard basis is implicit

In , when we write , we tacitly mean coordinates with respect to the standard basis :

When you change to a different basis, the same vector gets different coordinates.

32.2. Why coordinates matter

Coordinates make abstract vectors computable: once we fix a basis, every vector is a column of numbers and every linear transformation is a matrix.

Different bases reveal different aspects:

• Standard basis: most natural for
• Eigenbasis: makes the matrix of a linear transformation diagonal (when one exists — see Diagonalization)
• Orthonormal basis: distances and angles match the standard formulas
• Adapted basis: chosen to simplify a specific subspace structure

32.3. Connection to change of basis

When you switch between two bases and , the coordinate vectors are related by an invertible matrix — see Change of Basis.

32.4. See also

• Basis
• Change of Basis
• Vector Space
• Matrix Representation — coordinates of in terms of coordinates of