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Chapter Three: Maps Between Spaces. 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.
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Chapter Three: Maps Between Spaces • 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
3.I. Isomorphisms 3.I.1. Definition and Examples 3.I.2. Dimension Characterizes Isomorphism
3.I.1. Definition and Examples Definition 1.3: Isomorphism An isomorphismbetween 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 1.1: n-wide Row Vectors n-tall Column Vectors Example 1.2: PnRn+1 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(v1 + v2) = f(v1) + f(v2) and f(cv) = c f(v) (2) f preserves linear combinations of two vectors f(c1v1 + c2v2) = c1 f(v1) + c2 f(v2) (3) f preserves linear combinations of any finite number of vectors f(c1v1 +…+ cnvn) = c1 f(v1) +…+ cnf(vn) Proof: See Hefferon p.175.
Exercises 3.I.1 1. Show that the map f : R →R given by f(x) = x3 is one-to-one and onto. Is it an isomorphism? 2. (a) Show that a function f : R2 →R2 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 R2 with and Find
3. Show that, although R2 is not itself a subspace of R3, it is isomorphic to the xy-plane subspace of R3. 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 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 P2R2. (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 sumof 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.
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 Rn. Decomposition is unique for given B. Isomorphism classes are characterizedby dimension. Corollary 2.6: A finite-dimensional vector space is isomorphic to one and only one of the Rn.
Example 2.7: M22R4 where
Exercises 3.I.2. 1. Consider the isomorphism RepB(·) : P1 →R2 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 = V1V2 and that V is isomorphic to the space U under the map f. Show that U = f(V1) f(V2).
