2.5 Change of Bases 2.6 Dual Spaces Math 4377/6308 Advanced Linear Algebra 2.5 Change of Bases & 2.6 Dual Spaces Jiwen He Department of Mathematics, University of Houston [email protected] math.uh.edu/∼jiwenhe/math4377 Jiwen He, University of Houston Math 4377/6308, Advanced Linear Algebra Spring, 2015 1 / 14 2.5 Change of Bases 2.6 Dual Spaces 2.5 Change of bases Change of Coordinate Matrices. Similar Matrices. Jiwen He, University of Houston Math 4377/6308, Advanced Linear Algebra Spring, 2015 2 / 14 2.5 Change of Bases 2.6 Dual Spaces The Change of Coordinate Matrix Theorem (2.22) Let β and β0 be ordered bases for a finite-dimensional vector space β V , and let Q = [IV ]β0 . Then (a) Q is invertible. (b) For any v 2 V , [v]β = Q[v]β0 . Jiwen He, University of Houston Math 4377/6308, Advanced Linear Algebra Spring, 2015 3 / 14 2.5 Change of Bases 2.6 Dual Spaces The Change of Coordinate Matrix (cont.) β Q = [IV ]β0 is called a change of coordinate matrix, and we say that Q changes β0-coordinates into β-coordinates. Note that if Q changes from β0 into β coordinates, then Q−1 changes from β into β0 coordinates. Jiwen He, University of Houston Math 4377/6308, Advanced Linear Algebra Spring, 2015 4 / 14 2.5 Change of Bases 2.6 Dual Spaces Linear Operators A linear operator is a linear transformation from a vector space V into itself. Theorem Let T be a linear operator on a finite-dimensional vector space V with ordered bases β, β0. If Q is the change of coordinate matrix from β0 into β-coordinates, then −1 [T ]β0 = Q [T ]βQ Jiwen He, University of Houston Math 4377/6308, Advanced Linear Algebra Spring, 2015 5 / 14 2.5 Change of Bases 2.6 Dual Spaces Linear Operators (cont.) Corollary n Let A 2 Mn×n(F ), and γ an ordered basis for F . Then −1 [LA]γ = Q AQ, where Q is the n × n matrix with the vectors in γ as column vectors. Definition For A, B 2 Mn×n(F ), B is similar to A if the exists an invertible matrix Q such that B = Q−1AQ. Jiwen He, University of Houston Math 4377/6308, Advanced Linear Algebra Spring, 2015 6 / 14 2.5 Change of Bases 2.6 Dual Spaces Dual Spaces 2.6 Dual Spaces Dual Spaces and Dual Bases Transposes Jiwen He, University of Houston Math 4377/6308, Advanced Linear Algebra Spring, 2015 7 / 14 2.5 Change of Bases 2.6 Dual Spaces Dual Spaces Linear Functionals A linear functional on a vector space V is a linear transformation from V into its field of scalars F . Example Let V be the continuous real-valued functions on [0; 2π]. For a fix g 2 V , a linear functional h : V ! R is given by 1 Z 2π h(x) = x(t)g(t)dt 2π 0 Example Let V = Mn×n(F ), then f : V ! F with f (A) = tr(A) is a linear functional. Jiwen He, University of Houston Math 4377/6308, Advanced Linear Algebra Spring, 2015 8 / 14 2.5 Change of Bases 2.6 Dual Spaces Dual Spaces Coordinate Functions Example Let β = fx1; ··· ; xng be a basis for a finite-dimensional vector space V . Define fi (x) = ai , where 0 1 a1 B . C [x]β = @ . A an is the coordinate vector of x relative to β. Then fi is a linear functional on V called the ith coordinate function with respect to the basis β. Note that fi (xj ) = δij . Jiwen He, University of Houston Math 4377/6308, Advanced Linear Algebra Spring, 2015 9 / 14 2.5 Change of Bases 2.6 Dual Spaces Dual Spaces Dual Spaces Definition For a vector space V over F , the dual space of V is the vector space V ∗ = L(V ; F ). Note that for finite-dimensional V , dim(V ∗) = dim(L(V ; F )) = dim(V ) · dim(F ) = dim(V ) so V and V ∗ are isomorphic. Also, the double dual V ∗∗ of V is the dual of V ∗. Jiwen He, University of Houston Math 4377/6308, Advanced Linear Algebra Spring, 2015 10 / 14 2.5 Change of Bases 2.6 Dual Spaces Dual Spaces Dual Bases Theorem (2.24) Let β = fx1; ··· ; xng be an ordered basis for finite-dimensional vector space V , and let fi be the ith coordinate function w.r.t. β, ∗ ∗ ∗ and β = ff1; ··· ; fng. Then β is an ordered basis for V and for f 2 V ∗, n X f = f (xi )fi : i=1 Definition ∗ ∗ The ordered basis β = ff1; ··· ; fng of V that satisfies fi (xj ) = δij is called the dual basis of β. Jiwen He, University of Houston Math 4377/6308, Advanced Linear Algebra Spring, 2015 11 / 14 2.5 Change of Bases 2.6 Dual Spaces Dual Spaces Dual Bases (cont.) Theorem (2.25) Let V , W be finite-dimensional vector spaces over F with ordered bases β, γ. For any linear T : V ! W , the mapping T t : W ∗ ! V ∗ defined by T t (g) = gT for all g 2 W ∗ is linear t β∗ γ t with the property [T ]γ∗ = ([T ]β) . Jiwen He, University of Houston Math 4377/6308, Advanced Linear Algebra Spring, 2015 12 / 14 2.5 Change of Bases 2.6 Dual Spaces Dual Spaces Double Dual Isomorphism For a vector x 2 V , definex ^ : V ∗ ! F byx ^(f ) = f (x) for every f 2 V ∗. Note thatx ^ is a linear functional on V ∗, sox ^ 2 V ∗∗. Lemma For finite-dimensional vector space V and x 2 V , ifx ^(f ) = 0 for all f 2 V ∗, then x = 0. Jiwen He, University of Houston Math 4377/6308, Advanced Linear Algebra Spring, 2015 13 / 14 2.5 Change of Bases 2.6 Dual Spaces Dual Spaces Double Dual Isomorphism (cont.) Theorem (2.26) Let V be a finite-dimensional vector space, and define : V ! V ∗∗ by (x) =x ^. Then is an isomorphism. Corollary For finite-dimensional V with dual space V ∗, every ordered basis for V ∗ is the dual basis for some basis for V . Jiwen He, University of Houston Math 4377/6308, Advanced Linear Algebra Spring, 2015 14 / 14.