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-