# linear transformations as matrices

(3.4 hours to learn)

## Summary

Linear transformations from one vector space to another can be represented as matrices if a basis is given for each space. Common examples include projections, reflections, rotations, and scaling. Linear combinations, compositions and inverses of transformations can be analyzed in terms of matrix operations.

## Context

This concept has the prerequisites:

- dot product (Linear transformations are represented with matrix-vector products.)

## Core resources (read/watch one of the following)

## -Free-

→ Khan Academy: Linear Algebra

→ MIT Open Courseware: Linear Algebra (2011)

Videos for an introductory linear algebra course focusing on numerical methods.

Additional dependencies:

- bases
- vector spaces

## -Paid-

→ Multivariable Mathematics

A textbook on linear algebra and multivariable calculus with proofs.

Location:
Section 1.4, "Linear transformations and matrix algebra," pages 23-38

## Supplemental resources (the following are optional, but you may find them useful)

## -Free-

→ A First Course in Linear Algebra (2012)

A linear algebra textbook with proofs.

Location:
Section "Linear transformations"

Additional dependencies:

- complex vectors and matrices
- bases
- vector spaces

## -Paid-

→ Introduction to Linear Algebra

An introductory linear algebra textbook with an emphasis on numerical methods.

Location:
Section 7.1, "The idea of a linear transformation," pages 375-379, and Section 7.2, "The matrix of a linear transformation," up to "The identity transformation and the change of basis matrix," pages 384-390

Additional dependencies:

- bases
- matrix multiplication
- matrix inverse
- vector spaces

## See also

-No Additional Notes-