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### Test2

Course: MATH 3012, Fall 2011
School: Georgia Tech
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Word Count: 494

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Space Vector Axioms The kernel of T is {c Rn T (c) = 0} (a) Vector Addition (e) Additive Identity (b) Scalar Multiplication (f) Multiplicative Assoc. (c) Additive tivity (g) Multiplicative tity Commuta- (d) Additive Associativity Iden- b a Bases and Dimension 1 = (b) L2 = f 2 = a f (x) 2 dx b B is the basis for V if B spans V and B is linearly independent. f (x) dx (a) L = f (c) L =...

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Space Vector Axioms The kernel of T is {c Rn T (c) = 0} (a) Vector Addition (e) Additive Identity (b) Scalar Multiplication (f) Multiplicative Assoc. (c) Additive tivity (g) Multiplicative tity Commuta- (d) Additive Associativity Iden- b a Bases and Dimension 1 = (b) L2 = f 2 = a f (x) 2 dx b B is the basis for V if B spans V and B is linearly independent. f (x) dx (a) L = f (c) L = f You can have k linearly independent vectors in nspace, such that 0 k n. (Work in class and on hws from 3.76) (h) Proper Distribution L-Norms for C [a, b] 1 K + 1 vectors in K -space are linearly dependent. (Theorem 3.76) If B is independent then every vector x V can uniquely be represented by the elements of B . 12 = maxx[a,b] f (x) Inner Product for C [a, b] - < f, g >= b a f (x)g(x)dx Subspace With vector space V and a subset, S , of V , S is a subspace of V if S is also a vector space. To show this S must be closed under vector addition, scalar multiplication and be nonempty. (Theorem 3.14) If S and T are subspaces of a vector space of V then S T is a subspace of V . (Lemma 3.25) Spans Denition - The span of a set A is the set S of all the possible linear combinations of the elements of A. span{x1 , , xn } = {n=1 ck xk R} k Linear Denition Independency - A set of vectors is linearly independent if there is no nontrivial linear combination that equals zero. If S = {x1, , xn } is a set of nitely many vectors in V . And dene a function T Rn V, T (c) = n=1 ck xk Then T is injective IFF S is linearly indek pendent. The above T is surjective IFF S spans V . For nite B that is a basis for V , all other bases for V must have the same number of elements as B . V is a nite vector space. If {x1 , , xk } are independent, but dont span V , then there are vectors {xk+1 , , xn }so that B = {x1 , , xk , xk+1 , , xn } is a basis for V . If 2 are true all 3 are. B is independent, B spans V , and the number of vectors in B = dim(V ). B is a set of nitely many independent vectors in V . If no larger set contains B and is independent, then B is a basis for V . A minimal spanning set is a basis (ex. 3.94) Every spanning set contains a basis. (ex. 3.95) Denition - [x]B = (c1 , , cn ) that is, the components of x then make a component vector for x. Random things from class Pn is the set of all polynomials with degree at most n. F (R) is the set of all functions with real number coecients. The empty set is independent, but {0} is dependent. e1 = (1, 0, ), e2 = (0, 1, 0, ) form the standard basis for Rn
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