1. I. Homomorphisms & Isomorphisms II. Computing Linear Maps III. Matrix Operations VI. Change of Basis V. Projection Topics: Line of Best Fit Geometry of Linear Maps Markov Chains Orthonormal Matrices Chapter Three: Maps Between Vector Spaces

2. III.1. Homomorphisms & Isomorphisms Definition 1.1: Homomorphism & Isomorphism A map that preserves algebra structure is called a homomorphism. It is an isomorphism if the map is a bijection. It is an automorphism if the domain & codomain of the map are the same. Example: Real Vector Spaces h : V n → W m is a homomorphism if It is an isomorphism (denoted V  W ) if h is also 1-1 onto. A homomorphism between vectors spaces is also called a linear transformation, or linear map. Note:

3. Example 1.1:n-wide Row Vectors  n-tall Column Vectors Example 1.2: P n  R n+1 Example 1.4: Example 1.5: Example 1.6: Automorphisms : Rotation : Reflection : Example 1.7: Translation Dilation :

4. Theorem 2.1:Isomorphism is an equivalence relation between vector spaces. Proof: ( For details, see Hefferon p.179 ) 1) Reflexivity: V  V( h = id ) 2) Symmetry: V  W → W  V ( h  1 exists ) 3) Transitivity: V  W, W  U → V  U( h = f  g ) Theorem: V n  R n since Isomorphism classes are characterized by dimension.

5. Example 2.7: M 2  2  R 4 where

6. Exercises 3.I.1 1. Show that the map f : R → R given by f(x) = x 3 is one-to-one and onto. Is it an isomorphism? 2. (a) Show that a function f : R 2 → R 2 is an automorphism iff it has the form where a, b, c, d  R and ad  bc  0 (b) Let f be an automorphism of R 2 with and Find

7. 3. Consider the isomorphism Rep B (·) : P 1 → R 2 where B =  1, 1+x . Find the image of each of these elements of the domain. (a) 3  2x; (b) 2 + 2x; (c) x 4. Suppose that V = V 1  V 2 and that V is isomorphic to the space U under the map f. Show that U = f(V 1 )  f(V 2 ).