Today we start to study the linear transformation which is a big topic and an overview is given below. We have covered (1)-(4). The concepts of kernel and image of a linear transformation is a generalization of the nullspace and the column space of a matrix . In case you haven’t learnt nullspaces and column spaces before, please have some preparation — read Example 5.4.3 in p.258 (see p. 230, Def 5.10 for the definitions and Section 1.1-1.2 for elementary row operations and row echelon form).

- Definition (and examples) of linear transformations
- Linear transformation (from to ) induced by a matrix
- Basic properties of a linear transformation (Thm 7.1.1)
- Construction of a linear transformation (Thm 7.1.3)
- The kernel and image of a linear transformation : subspace property (Thm 7.2.1), characterization of one-to-one linear transformation (Thm 7.2.2), Dimension Theorem (Thm 7.2.4)
- Isomorphism and isomorphic vector spaces: Definition (7.4), basic properties (Thm 7.3.1), properties of isomorphic vector spaces (Thm 7.3.2 and its corollaries)
- Composition of linear transformations: when the inverse of a linear transformation exists (Thm 7.3.5)
- Standard matrix representations of linear transformations from to
- Coordinates of a vector space (p.350 and Def 9.1, Example 7.3.7, Thm 9.1.1)
- Change of bases for coordinates (Def 9.4, Thm 9.22)
- Matrix representations of linear transformations (Thm 9.1.2, Def 9.2): properties (Thm 9.1.3-5)
- Change of bases for matrix representations
- Operators and Similarity (Section 9.2)

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