Topics: Algebra - Linear Transformation

Given a linear transformation , the associated matrix of is a matrix that represents on the bases of and .

This matrix can be used to transform elements by having it multiply the respective coordinate vector.

Operated and composed transformations can also have an associated matrix.



  • vector spaces with finite dimension
  • a basis of
  • a basis of
  • A linear transformation

The associated matrix is a matrix that contains elements such that:

Such a matrix is denoted by also , and is also known as a basis change matrix (matriz de cambio de base).


To build the associated matrix of a linear transformation , we can follow three steps:

  1. Transform each one of the elements in
  2. Under , find the coordinate vectors of the transformed elements
  3. Build the matrix by using these coordinate vectors as its columns, in order.

Determinant of Associated Matrices


Let . Let and be bases for . of under different bases. We have that: