This chapter introduces vectors and vector spaces. The vector spaces R1, R2, and R3 are focused on but a discussion of abstract vector spaces is included in the end. It includes visualizations of vectors in R1, R2, and R3.

This chapter introduces the operations of vector addition and scaling. The algebraic aspect of the operation is covered but an emphasis is placed on a visual understanding through the use of the tip to tail method and interactive visualizations.

This chapter introduces the concept of a linear combinations as the composition of vector addition and scaling. Visualize the linear combination of up to three vectors with the interactive linear combination visualizer.

This chapter introduces the concept of span and its relation to linear dependence. Visualize the span of up to three vectors with the interactive span visualizer.

This chapter introduces the concept of a basis for a vector space and what a coordinate vector is. Use the interactive program to visualize and create a custom basis for R3.

This chapter covers linear transformations and how matrices can encode a change of basis. The geometric view of matrices is emphasized with three visualizations showing how matrices can encode shearing, rotation and scaling.