9: Vector Spaces
( \newcommand{\kernel}{\mathrm{null}\,}\)
- 9.1: Algebraic Considerations
- In this section we consider the idea of an abstract vector space.
- 9.2: Spanning Sets
- In this section we will examine the concept of spanning introduced earlier in terms of Rn . Here, we will discuss these concepts in terms of abstract vector spaces.
- 9.3: Linear Independence
- In this section, we will again explore concepts introduced earlier in terms of Rn and extend them to apply to abstract vector spaces.
- 9.4: Subspaces and Basis
- In this section we will examine the concept of subspaces introduced earlier in terms of Rn. Here, we will discuss these concepts in terms of abstract vector spaces.
- 9.8: The Kernel and Image of a Linear Map
- Here we consider the case where the linear map is not necessarily an isomorphism. First here is a definition of what is meant by the image and kernel of a linear transformation.
- 9.9: The Matrix of a Linear Transformation
- You may recall from Rn that the matrix of a linear transformation depends on the bases chosen. This concept is explored in this section, where the linear transformation now maps from one arbitrary vector space to another.