# Linear Algebra II

## September 22, 2014

### Linear Transformation – A little overview

Filed under: 2014 Fall — Y.K. Lau @ 7:24 PM

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 ${{\rm null}\, A}$ and the column space ${{\rm col}\, A}$ of a matrix ${A}$. 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).

1. Definition (and examples) of linear transformations
2. Linear transformation (from ${{\mathbb R}^n}$ to ${{\mathbb R}^m}$) induced by a matrix ${A\in M_{m,n}}$
3. Basic properties of a linear transformation (Thm 7.1.1)
4. Construction of a linear transformation (Thm 7.1.3)
5. The kernel ${\ker T}$ and image ${{\rm im}\, T}$ of a linear transformation ${T}$: subspace property (Thm 7.2.1), characterization of one-to-one linear transformation (Thm 7.2.2), Dimension Theorem (Thm 7.2.4)
6. 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)
7. Composition of linear transformations: when the inverse of a linear transformation exists (Thm 7.3.5)
8. Standard matrix representations of linear transformations from ${{\mathbb R}^n}$ to ${{\mathbb R}^m}$
9. Coordinates of a vector space (p.350 and Def 9.1, Example 7.3.7, Thm 9.1.1)
10. Change of bases for coordinates (Def 9.4, Thm 9.22)
11. Matrix representations of linear transformations (Thm 9.1.2, Def 9.2): properties (Thm 9.1.3-5)
12. Change of bases for matrix representations
13. Operators and Similarity (Section 9.2)