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

3.I. Isomorphisms 3.I.1. Definition and Examples 3.I.2. Dimension Characterizes Isomorphism

3.I.1. Definition and Examples Example 1.1:n-wide Row Vectors  n-tall Column Vectors Example 1.2: P n  R n+1 Definition 1.3: Isomorphism An isomorphism between two vector spaces V and W is a map f : V →W that (1) is a correspondence: f is a bijection (1-to-1 and onto); (2) preserves structure: In which case, V  W, read “ V is isomorphic to W. ” Example 1.4: Example 1.5:

Automorphism Automorphism = Isomorphism of a space with itself. Example 1.6: Dilation : Rotation : Reflection : Example 1.7: Translation Symmetry: Invariance under mapping.

Lemma 1.8:An isomorphism maps a zero vector to a zero vector. Proof: Lemma 1.9: For any map f : V → W between vector spaces these statements are equivalent. (1) f preserves structure f(v 1 + v 2 ) = f(v 1 ) + f(v 2 ) and f(cv) = c f(v) (2) f preserves linear combinations of two vectors f(c 1 v 1 + c 2 v 2 ) = c 1 f(v 1 ) + c 2 f(v 2 ) (3) f preserves linear combinations of any finite number of vectors f(c 1 v 1 +…+ c n v n ) = c 1 f(v 1 ) +…+ c n f(v n ) Proof: See Hefferon p.175.

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

3. Show that, although R 2 is not itself a subspace of R 3, it is isomorphic to the xy-plane subspace of R 3. This is a vector space, the external direct sum (Cartesian product) of U and W. (a) Check that it is a vector space. (b) Find a basis for, and the dimension of, the external direct sum P 2  R 2. (c) What is the relationship among dim(U), dim(W), and dim(U  W)? (d) Suppose that U and W are subspaces of a vector space V such that V = U  W (in this case we say that V is the internal direct sum of U and W). Show that the map f : U  W → V given by ( u, w )  u + w is an isomorphism. Thus if the internal direct sum is defined then the internal and external direct sums are isomorphic. 4. Let U and W be vector spaces. Define a new vector space, consisting of the set U  W = { ( u, w ) | u  U and w  W } along with these operations. and

3.I.2. Dimension Characterizes Isomorphism Theorem 2.1:Isomorphism is an equivalence relation between vector spaces. Proof: ( For details, see Hefferon p.179 ) 1) Reflexivity: Identity map, id: v  v, preserves L.C. 2) Symmetry: f is bijection → f  1 exists & preserves L.C. 3) Transitivity: Composition preserves L.C. Isomorphism classes:

Theorem 2.3: Vector spaces are isomorphic  they have the same dim. Proof: (see Hefferon p.180) Isomorphism → correspondence between bases. Lemma 2.4:If spaces have the same dimension then they are isomorphic. Proof: (see Hefferon p.180) Every n-D vector space is isomorphic to R n. Isomorphism classes are characterized by dimension. Corollary 2.6: A finite-dimensional vector space is isomorphic to one and only one of the R n. is unique for given B. Decomposition

Example 2.7: M 2  2  R 4 where

Exercises 3.I 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 2. 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 ).