MAT 531 Geometry/Topology II Introduction to Smooth Manifolds Claude LeBrun Stony Brook University April 9, 2020 1 Dual of a vector space: 2 Dual of a vector space: Let V be a real, finite-dimensional vector space. 3 Dual of a vector space: Let V be a real, finite-dimensional vector space. Then the dual vector space of V is defined to be 4 Dual of a vector space: Let V be a real, finite-dimensional vector space. Then the dual vector space of V is defined to be ∗ V := fLinear maps V ! Rg: 5 Dual of a vector space: Let V be a real, finite-dimensional vector space. Then the dual vector space of V is defined to be ∗ V := fLinear maps V ! Rg: ∗ Proposition. V is finite-dimensional vector space, too, and 6 Dual of a vector space: Let V be a real, finite-dimensional vector space. Then the dual vector space of V is defined to be ∗ V := fLinear maps V ! Rg: ∗ Proposition. V is finite-dimensional vector space, too, and ∗ dimV = dimV: 7 Dual of a vector space: Let V be a real, finite-dimensional vector space. Then the dual vector space of V is defined to be ∗ V := fLinear maps V ! Rg: ∗ Proposition. V is finite-dimensional vector space, too, and ∗ dimV = dimV: ∗ ∼ In particular, V = V as vector spaces. 8 Dual of a vector space: Let V be a real, finite-dimensional vector space. Then the dual vector space of V is defined to be ∗ V := fLinear maps V ! Rg: ∗ Proposition. V is finite-dimensional vector space, too, and ∗ dimV = dimV: ∗ ∼ In particular, V = V as vector spaces. But there is no natural, preferred isomorphism! 9 Dual of a vector space: Let V be a real, finite-dimensional vector space. Then the dual vector space of V is defined to be ∗ V := fLinear maps V ! Rg: ∗ Proposition. V is finite-dimensional vector space, too, and ∗ dimV = dimV: ∗ ∼ In particular, V = V as vector spaces. But there is no natural, preferred isomorphism! Let's not confuse one twin for the other! 10 In fact, 9 contravariant functor from category 11 In fact, 9 contravariant functor from category (fvector spaces over Rg; fR-linear mapsg) 12 In fact, 9 contravariant functor from category (fvector spaces over Rg; fR-linear mapsg) to itself, given by ∗. V V 13 In fact, 9 contravariant functor from category (fvector spaces over Rg; fR-linear mapsg) ∗ to itself, given by V V . Namely, if A : V ! W any linear map, 14 In fact, 9 contravariant functor from category (fvector spaces over Rg; fR-linear mapsg) ∗ to itself, given by V V . Namely, if A : V ! W any linear map, ∗ ∗ ∗ 9 adjoint map A : W ! V , defined by 15 In fact, 9 contravariant functor from category (fvector spaces over Rg; fR-linear mapsg) ∗ to itself, given by V V . Namely, if A : V ! W any linear map, ∗ ∗ ∗ 9 adjoint map A : W ! V , defined by (A∗')(v) = '(A(v)): 16 In fact, 9 contravariant functor from category (fvector spaces over Rg; fR-linear mapsg) ∗ to itself, given by V V . Namely, if A : V ! W any linear map, ∗ ∗ ∗ 9 adjoint map A : W ! V , defined by (A∗')(v) = '(A(v)): ∗ ∗ A ∗ V − W A V −! W 17 In fact, 9 contravariant functor from category (fvector spaces over Rg; fR-linear mapsg) ∗ to itself, given by V V . Namely, if A : V ! W any linear map, ∗ ∗ ∗ 9 adjoint map A : W ! V , defined by (A∗')(v) = '(A(v)): ∗ ∗ A ∗ V − W A V −! W &# R 18 ∼ n Example. Let V = R be the set of column vectors of length n: 19 ∼ n Example. Let V = R be the set of column vectors of length n: 8 2 a1 3 9 > > < 6 a2 7 = V = 6 . 7 > 4 . 5 > :> an ;> 20 ∼ n Example. Let V = R be the set of column vectors of length n: 8 2 a1 3 9 > > < 6 a2 7 = V = 6 . 7 > 4 . 5 > :> an ;> ∗ ∼ n Then V = R can be identified with the set of row vectors of length n 21 ∼ n Example. Let V = R be the set of column vectors of length n: 8 2 a1 3 9 > > < 6 a2 7 = V = 6 . 7 > 4 . 5 > :> an ;> ∗ ∼ n Then V = R can be identified with the set of row vectors of length n ∗ V = b1 b2 ··· bn 22 ∼ n Example. Let V = R be the set of column vectors of length n: 8 2 a1 3 9 > > < 6 a2 7 = V = 6 . 7 > 4 . 5 > :> an ;> ∗ ∼ n Then V = R can be identified with the set of row vectors of length n ∗ V = b1 b2 ··· bn because any linear map V ! R takes the form 23 ∼ n Example. Let V = R be the set of column vectors of length n: 8 2 a1 3 9 > > < 6 a2 7 = V = 6 . 7 > 4 . 5 > :> an ;> ∗ ∼ n Then V = R can be identified with the set of row vectors of length n ∗ V = b1 b2 ··· bn because any linear map V ! R takes the form 2 a1 3 2 a1 3 2 2 6 a 7 6 a 7 ~ 6 7 7−! b1 b2 ··· bn 6 7 9!b: 4 . 5 4 . 5 an an 24 ∼ n Example. Let V = R be the set of column vectors of length n: 8 2 a1 3 9 > > < 6 a2 7 = V = 6 . 7 > 4 . 5 > :> an ;> ∗ ∼ n Then V = R can be identified with the set of row vectors of length n ∗ V = b1 b2 ··· bn because any linear map V ! R takes the form 2 a1 3 2 a1 3 2 2 6 a 7 6 a 7 ~ 6 7 7−! b1 b2 ··· bn 6 7 9!b: 4 . 5 4 . 5 an an 25 ∼ n ∼ m Example. If V = R , W = R , and 26 ∼ n ∼ m Example. If V = R , W = R , and A : V ! W 27 ∼ n ∼ m Example. If V = R , W = R , and A : V ! W given by left-multiplication 28 ∼ n ∼ m Example. If V = R , W = R , and A : V ! W given by left-multiplication 2 a1 3 2 a1 3 2 A ··· A 3 a2 11 1n a2 6 7 7−! . 6 7 6 . 7 4 5 6 . 7 4 5 A ··· A 4 5 an m1 mn an 29 ∼ n ∼ m Example. If V = R , W = R , and A : V ! W given by left-multiplication 2 a1 3 2 a1 3 6 a2 7 6 a2 7 6 7 7−! 6 7 4 . 5 A 4 . 5 an an 30 ∼ n ∼ m Example. If V = R , W = R , and A : V ! W given by left-multiplication 2 a1 3 2 a1 3 2 A ··· A 3 a2 11 1n a2 6 7 7−! . 6 7 6 . 7 4 5 6 . 7 4 5 A ··· A 4 5 an m1 mn an 31 ∼ n ∼ m Example. If V = R , W = R , and A : V ! W given by left-multiplication 2 a1 3 2 a1 3 2 A ··· A 3 a2 11 1n a2 6 7 7−! . 6 7 6 . 7 4 5 6 . 7 4 5 A ··· A 4 5 an m1 mn an then ∗ ∗ ∗ A : W ! V 32 ∼ n ∼ m Example. If V = R , W = R , and A : V ! W given by left-multiplication 2 a1 3 2 a1 3 2 A ··· A 3 a2 11 1n a2 6 7 7−! . 6 7 6 . 7 4 5 6 . 7 4 5 A ··· A 4 5 an m1 mn an then ∗ ∗ ∗ A : W ! V given by right-multiplication 33 ∼ n ∼ m Example. If V = R , W = R , and A : V ! W given by left-multiplication 2 a1 3 2 a1 3 2 A ··· A 3 a2 11 1n a2 6 7 7−! . 6 7 6 . 7 4 5 6 . 7 4 5 A ··· A 4 5 an m1 mn an then ∗ ∗ ∗ A : W ! V given by right-multiplication 2 3 A11 ··· A1n . b1 b2 ··· bm 7−! b1 b2 ··· bm 4 . 5 Am1 ··· Amn 34 ∼ n ∼ m Example. If V = R , W = R , and A : V ! W given by left-multiplication 2 a1 3 2 a1 3 6 a2 7 6 a2 7 6 7 7−! 6 7 4 . 5 A 4 . 5 an an then ∗ ∗ ∗ A : W ! V given by right-multiplication b1 b2 ··· bm 7−! b1 b2 ··· bm A 35 Example. If we \artificially” identify column vec- tors with row vectors 2 3 b1 t 6 b2 7 b1 b2 ··· bm = 6 . 7 4 . 5 bm then the action of A∗ becomes 2 3 2 3 b1 2 3 b1 A11 ··· Am1 6 b2 7 . 6 b2 7 6 . 7 7! 4 . 5 6 . 7 4 . 5 4 . 5 A1n ··· Amn bm bm so adjoint A∗ sometimes called transpose of A. 36 Example. If we \artificially” identify column vec- tors with row vectors 2 3 b1 t 6 b2 7 b1 b2 ··· bm = 6 . 7 4 . 5 bm then the action of A∗ becomes 2 3 2 3 b1 b1 6 b2 7 t 6 b2 7 6 . 7 7! 6 . 7 4 . 5 A 4 . 5 bm bm so adjoint A∗ sometimes called transpose of A. 37 9 contravariant functor from category (fvector spaces over Rg; fR-linear mapsg) ∗ to itself given by V V . If A : V ! W any linear map, ∗ ∗ ∗ 9 adjoint map A : W ! V , defined by (A∗')(v) = '(A(v)): ∗ ∗ A ∗ V − W A V −! W 38 9 contravariant functor from category (fvector spaces over Rg; fR-linear mapsg) ∗ to itself given by V V . If B : V ! W any linear map, ∗ ∗ ∗ 9 adjoint map B : W ! V , defined by I∗ = I: ∗ I ∗ V − V I V −! V 39 9 contravariant functor from category (fvector spaces over Rg; fR-linear mapsg) ∗ to itself given by V V .