5: Vector Spaces and Subspaces
- Page ID
- 170466
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- In this section we consider the idea of an abstract vector space.
- 5.7: Spanning, Linear Independence and Basis in Rⁿ
- By generating all linear combinations of a set of vectors one can obtain various subsets of \(\mathbb{R}^{n}\) which we call subspaces. For example what set of vectors in \(\mathbb{R}^{3}\) generate the \(XY\)-plane? What is the smallest such set of vectors can you find? The tools of spanning, linear independence and basis are exactly what is needed to answer these and similar questions and are the focus of this section.
- 5.10: 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.
- 5.11: 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.