Linear Transformations

We already discusses how to abstract the our concrete examples of vectors as lists of numbers into the general notion of a vector space.

We saw that almost every object we have touched could be put in its own vector space and then studied with the tools of linear algebra.

Next, we abstract matrices. The key properties of matrices are:

• Multiplication by a matrix commutes with scaling \(A(cx) = cA(x)\)
• Multiplication by a matrix commutes with addition of vectors \(A(x + y) = A(x) + A(y)\)

We will use there two properties as the specification for the notion of a linear transformation of vectors spaces.

Table of contents

• Definitions and examples
• Linear Transformations in Bases
• Isomorphisms
• Null Spaces and Ranges
• Eigenvectors
• Determinants
• Cofactor Formula
• Volumes
• Characteristic Polynomials
• Diagonalization over $\mathbb{C}$
• Cayley-Hamilton
• Jordan Canonical Form