MATHEMATICS -NYC- Vectors and Matrices Martin Huard Fall 7 XV - Vector Spaces and Subspaces Describe the zero vector (the additive identity) for the following vector spaces 4 a) c) d) e) C, b) x, y x, y xx, y y k x, y x k, ky M, V x, y : x, y, x with the following operations : Describe the additive inverse of a vector for the following vector spaces a) c) d) 4 e) C, b) x, y x, y xx, y y k x, y x k, ky M, V x, y : x, y, x with the following operations : Determine whether the given set, together with the indicated operations, is a vector space If it is, prove that each axiom is satisfied, if it is not, identify the axioms that fail a) M, with standard operation b) with standard operation c) P with the standard operation x, y : x, y with standard operations d) The set e) The set of all matrices of the form f) The set ax : a g) h) i) j) a P P b with standard operations with the following operations : x, y x, y x x, y y k x, y kx, y with the following operations : x, y x, y x, k x, y kx, ky with the following operations : x, y x, y xx, y y k x, y kx, ky with the following operations : x, y x, y x x, y y,, k x y k x k y
4 Consider the set V whose only element is moon, that is, V moon Is this set a vector space under the following operations? moon moon = moon k (moon) = moon for every real number k Determine whether the subset of, with the standard operations, is a vector space Justify your answer (Hint: Show that is a subspace of ) a, b, : a, b a,, : a a) b) c) a, b, a b : a, b d) e) a, b a, b : a, b f) g) x, y, z : x y z h) x, y, z : x t, y t, z t, t i) u : u w,, 6 Determine whether the subset of Justify your answer a b a) : a, b, c, d c d b) is the set of matrices A such that det( A) c) is the set of symmetric matrices A d) is the set of diagonal matrices a b e) : a, b, c c f) a b : a b c d c d a, b, ab : a, b x, y, z : x y z with the standard operations is a vector space 7 Determine whether the subset of C, is a subspace of, answer a) The set of nonnegative functions: f x b) The set of all even functions: f x f x c) The set of all odd functions: f x f x d) The set of all constant functions: f x c, c e) The set of all functions such that f f) The set of all functions such that f C Justify your Fall 7 Martin Huard
ANSERS,,, a) e) (,) a) u u, u, u, u4 - u u, u, u, u d) a) Let 4 b) f( x) c) p x a a x a x a x p x a a x a x a x b) f x f x c) e) u x, y -u x, y a a a b b b c c c A, B and C a a a b b b c c c A a b a b a b is a matrix B a b a b a b d) p( x) x x x a a a A a a a a a a -A a a a a b a b a b b a b a b a A B B A a b a b a b b a b a b a a a a b c b c b c a a a b c b c b c A B C 4 A a b c a b c a b c a b c a b c a b c a b c a b c a b c a b c a b c a b c = a b a b a b c c c = a b a b a b c c c = A B C a a a a a a a a a a a a a a a a a a A Fall 7 Martin Huard
a a a a a a a a a a a a A A 6 a a a a a a a a a a a a ka ka ka ka 7 k A B ka ka ka is a matrix k a b k a b k a b k a b k a b k a b ka kb ka kb ka kb ka kb ka ka ka ka ka kb k la k la k la k l A k la k la k la ka la ka la ka la ka la ka la ka la ka la k la k la k la k la k la k la k la 8 9 kl A kl a kl a kl a kl a kl a kl a a a a a a a A A a a a a a a = b) Let u u, u, u, v v, v, v and w w, w, w u v u v, u v, u v u v u v, u v, u v v u, v u, v u v u u v w u v v, u v w, u v w u v w, u v w, u v w = u v w 4 u u, u, u,, u, u, u u, u, u u u +- u u u u u u u u u u u u u,,,,,,,, Fall 7 Martin Huard 4
c) Let 6 ku ku, ku, ku 7 k u v k u v, u v, u v k u v, k u v, k u v ku kv, ku kv, ku kv ku kv k l u k l u, k l u, k l u 8 ku lu, ku lu, ku lu ku lu k lu k lu, lu, lu kl u, kl u, kl u kl u 9 u u, u, u u, u, u = u p ( x) a x a x a x a r ( x) c x c x c x c q ( x) b x b x b x b and ( x) ( x) a b x a b x a b x a b p q is a rd degree polynomial p( x) q( x) a b x a b x a b x a b b a x b a x b a x b a q( x) p( x) p( x) q( x) r( x) ax ax ax a b c x b c x b c x b c a b c x a b c x a b c x a b c a b c x a b c x a b c x a b c = a b x a b x a b x a b cx cx cx c = p( x) q( x) r( x) 4 p( x) a x a x a x a a x a x a x a p( x) p ( x) + -p( x) a a x a a x a a x a a x x x kp ( x) ka x ka x ka x ka is a rd degree polynomial 6 Fall 7 Martin Huard
7 k p( x) q( x) k a b x a b x a b x a b ka kb x ka kb x ka kb x ka kb ka x ka x ka x ka kb x kb x kb x kb kp( x) kq( x) 8 p( ) k l x k l a x k l a x k l a x k l a kax kax ka x ka lax lax lax la kp( x) lp( x) 9 k lp( x) k la x la x la x la kla x kla x kla x kla kl p ( x) p ( x) a x a x a x a a x a x a x a = p ( x) d) Axiom is not satisfied, there is no, -u - u u u u is not in the set Axiom 6 is not satisfied because if being < k u+- u because in the set such that then ku ku, ku is not in the set, ku and ku a b a b e) Axiom is not satisfied since A B a b a b in the set Axiom 4 is not satisfied since is not in the set a Axiom is not satisfied since - A a is not in the set Axiom 6 in not satisfied since ka ka k k ka is not in the set if k is not f) Let p, ( x) ax q and ( x) bx r ( x) cx p( x) q ( x) ax bx a b x is in the set p( x) q( x) a b x b a x q( x) p ( x) p( ) q( ) r( ) 4 x x x ax b c x a b c x a b c x p( x) a x ax p ( x) p ( x) + -p( x) a a x x 6 kp ( x) kax is in the set 7 a b x cx = p( x) q( x) r( x) k p x q x k a b x ka kb x kax kbx kp x kq x ( ) ( ) ( ) ( ) Fall 7 Martin Huard 6
8 k l p( x) k l ax kax lax kp( x) lp ( x) 9 k lp( x) k lax klax kl p ( x) p ( x) ax ax = p ( x) g) The set is not a vector space since axiom 8 fails For example, let k, l and k l u,,, k u l u,,,,, Thus k l u k u l u Axioms 4 and also fail h) The set is not a vector space because axiom fails For example, let u, v,,,,,,, u v v u Thus u v v u Axioms 4, and 8 also fail u u, u, u if u i) Axiom 4 fails since Axiom and 7 also fail j) Axiom 8 fails,, k l u k l u k l u k l u k u klu l u, k u klu l u, k u klu l u k u kl u l u Axiom also fails k u l u 4 Yes It is similar to the vector space V a) Yes,,,,,, ku k u, u, ku, ku, u v u u v v u v u v u v u,, v,, u v,, b) No c) Yes,,,,,, ku k u, u, u u ku, ku, ku ku u v u u u u v v v v u v u v u v u v ku k u, u, u u ku, ku, ku u d) No since k u, and ku ku k u u ku u if Fall 7 Martin Huard 7
e) Yes ku k u, u u, u ku, ku ku, u u v u, u u, u v, v v, v u v, u v u v, u v f) Yes If u then u u u and if v then v v v u v u, u, u v, v, v u v, u v, u vv Since u v u v u v u u u v v v then u v ku k u, u, u ku, ku, ku Since ku ku ku k u u u k then ku g) No If u then u u u and if v then v v v u v u, u, u v, v, v u v, u v, u vv Since u v u v u v u u u v v v 6 then u v h) Yes If u u t, t,t and if v then then v s, s,s u v t, t,t s, s,s,,,, ku k t, t,t kt, kt,kt kt, kt,kt t s t s t s t s t s t s i) Yes If u, v, then uw and vw u v since u v w u w v w since ku w k u w k ku 6 a) No e do not always have closure under scalar multiplication For a example if k and A=, then b) No Let A= and B Then det A det B so A, B Since det A B, then AB, so we do not have closure under addition T T c) Yes Let A and B be symmetric matrices, A A and B B T T T A B A B A B, hence AB Fall 7 Martin Huard 8
T T ka ka ka, hence ka Hence is a subspace of d) Yes Let A= a a and b B b be in a b A B ka ka ka Hence is a subspace of e) Yes a b a a b b a b a b A B a b a b a a ka ka ka k a ka Hence is a subspace of f) Yes a a b b a b a b A B a a b b a b a b since a b a b a b a b a a a a b b b b a a ka ka ka k a a ka ka since ka ka ka ka k a a a a Hence is a subspace of 7 a) No e do not always have closure under scalar multiplication For a example if k, then f ( x) since f ( x) b) Yes f g x since f gx f ( x) g( x) f ( x) g( x) f g x kf ( x) since kf ( x) kf ( x) kf x kf x c) Yes f g x since f gx f ( x) g( x) f ( x) g( x) f g x kf ( x) since kf ( x) kf ( x) kf x kf x d) Yes f g x since f g x f ( x) g ( x) c d is a constant kf ( x) since kf( x) kf( x) kc is a constant e) Yes f g x since f g f () g () kf ( x) since kf() kf () k f) No f g x since f g f () g () kf ( x) since kf () k k if k Fall 7 Martin Huard 9