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Section_6_Vectors_spaces_and_subspaces

Section_6_Vectors_spaces_and_subspaces - Section 6 Vector...

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Section 6 Vector spaces and subspaces Problem 1 Define the following concepts : a) S is a subspace b) The subspace S = span{ w v u , , } c) Linear independence of vectors w v u , , . d) The vectors w v u , , are a basis for a subspace S. Problem 2 Determine whether or not the following are subspaces. a) From R 3 , let S a be the set of all vectors whose length is 1. b) From R 2x2 , let S b be the set of all 2 x 2 matrices whose sum of all entries equal 0. c) From P 3 , let S c be the set of all 3 rd degree polynomials such that p(0) = 1. Solution a) not a subspace b) subspace c) not a subspace Problem 3 Consider the matrices 1 1 1 1 1 0 3 1 2 1 0 1 3 2 1 M M M a) Show that the matrices M 1 , M 2 and M 3 are linearly independent. b) Express the matrix 3 5 5 6 M as a linear combination of M 1 , M 2 and M 3 . c) Find a new set of linearly independent matrices N 1 , N 2 and N 3 such that span{ N 1 , N 2 , N 3 } = span{ M 1 , M 2 , M 3 }. Solutions a) The only solution to aM 1 + bM 2 + c M 3 = 0 is a = 0, b = 0, c = 0. b) We want to obtain a, b, c such that aM 1 + bM 2 + c M 3 = M
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1 0 1 3 1 1 6 5 1 2 0 1 1 1 5 3 3 6 5 2 5 3 a b c a b c b c a c a b c Solving for a, b and c, we find a = 3, b = 1, c = 2 c) Replace any matrix by a scalar multiple, or by a combination of two of them… or even better reduce the basis vectors to their simplest form: 1 0 1 2 1 0 1 2 1 0 1 2 1 3 0 1 0 3 1 3 0 1 0 3 1 1 1 1 0 1 0 3 0 3 1 3 1 0 1 2 1 0 0 8 0 1 0 3 0 1 0 3 0 0 1 6 0 0 1 6 1 2 3 1 0 0 1 0 0 0 8 0 3 1 6 N N N Problem 4 The set of all 2 nd degree polynomials, P , is a vector space under the following rules of addition and scalar multiplication : Let p( x ) = a 0 + a 1 x + a 2 x 2 and q( x ) = b 0 + b 1 x + b 2 x 2 (p + q)( x ) = (a 0 + b 0 )+ (a 1 + b 1 ) x + (a 2 + b 2 ) x 2 = p( x ) + q( x ); (kp)( x ) = ka 0 + ka 1 x + ka 2 x 2 = k[p( x )]. a) Show that S 1 , the set of all elements p ( x ) P such that p(2) = 0, is a subspace. b) Find a basis for the S 1 . State its dimension. c) Show that the polynomial p( x ) = 2 + 3 x 2 x 2 is an element of S 1 and express it as a combination of the basis for S 1 . d) Show that S 2 , the set of all elements p ( x ) P such that p’(1) = 0, is a subspace. e) Find a basis for the S 2 . State its dimension.
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