Matrix representation

Japanese school year: University year 2

What you learn

Learn how to represent linear mappings as matrices by choosing bases for vector spaces. This technique allows abstract linear transformations to be computed concretely using matrix algebra. Prior knowledge of the definition of linear maps and vector space bases is recommended.

Go to practice

Key points

This formula shows that a linear map $f$ on a vector $\boldsymbol{x}$ can be computed simply by multiplying by matrix $A$ . Once bases are chosen, abstract transformations become concrete matrix algebra.

f(\boldsymbol{x}) = A \boldsymbol{x}

This formula explains how to build the representation matrix $A$ . The coordinates $a_{ij}$ of each transformed basis vector $f(\boldsymbol{v}_j)$ along output basis $\boldsymbol{w}_i$ form column $j$ of the matrix.

f(\boldsymbol{v}_j) = \sum_{i=1}^m a_{ij} \boldsymbol{w}_i

This formula shows how a representation matrix transforms when changing coordinate bases. The new matrix $B$ is found by sandwiching the original matrix $A$ between transition matrix $P$ and its inverse $P^{-1}$ .

B = P^{-1} A P

Choose a set to practice.

Practice 1Coming soon