• Table of Contents

Linear Mappings

This section introduces some basic concepts describing a special class of functions between vector spaces.

Why? We want to formalize the idea of mappings on vector spaces that preserve its structure

• “Structure” refers to the rules which define a vector space (i.e. maps that maintain “vectoriness”)

Algebraic Definition

Injectivity, Surjectivity

Matrix Representation

Fundamental Subspaces

In this section, we associate a linear transformation $T: V \rightarrow W$ with 2 subspaces we call the image and kernel of $T$. These subspaces are important in determining properties of $T$ (the dimension theorem)

• We also establish 2 more important subspaces of a matrix: The row space, and the left null space.

Kernel/Null Space

Image/Column Space

Dimension Theorem

Rank

Row Space