Adv Alegbra HW Solutions 129

Adv Alegbra HW Solutions 129 - 129 (ii) If U and U are...

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Unformatted text preview: 129 (ii) If U and U are subspaces of a finite dimensional vector space V , prove that dim(U ) + dim(U ) = dim(U ∩ U ) + dim(U + U ). Solution. Take a basis of U ∩ U and extend it to bases of U and of U . (iii) If V is finite dimensional, prove that every subspace U of V is a direct summand. Solution. Extend a basis B of U to a basis B ∪ C of V , and show that V = U ⊕ C . 4.20 (i) Prove that U ⊕ W is a vector space. Solution. Absent. (ii) If U and W are finite dimensional vector spaces over a field k , prove that dim(U ⊕ W ) = dim(U ) + dim(W ). Solution. If B is a basis of U and C is a basis of W , then B ∪ C is a basis of U ⊕ V . 4.21 Assume that V is an n -dimensional vector space over a field k and that V has a nondegenerate inner product. If W is an r -dimensional subspace of V , prove that V = W ⊕ W ⊥ (see Example 4.5). Conclude that dim(W ⊥ ) = n − r. 4.22 Here is a theorem of Pappus holding in k 2 , where k is a field. Let and m be distinct lines, let A1 , A2 , A3 be distinct points on , and let B1 , B2 , B3 be distinct points on m . Define C1 to be A2 B3 ∩ A3 B2 , C2 to be A1 B3 ∩ A3 B1 , and C3 to be A1 B2 ∩ A2 B1 . Then C1 , C2 , C3 are collinear. State the dual of the theorem of Pappus. Solution. Let P and Q be distinct points, let 1 , 2 , 3 be distinct lines passing through P , and let m 1 , m 2 , m 3 be distinct liness passing through Q . Define n 1 to be the line determined by 2 ∩ m 3 and 3 ∩ m 2 , n 2 to be the line determined by 1 ∩ m 3 and 3 ∩ m 1 , and n 3 to be the line determined by 1 ∩ m 2 and 2 ∩ m 1 . Then n 1 , n 2 , n 3 intersect in a point. 4.23 True or false with reasons. (i) There is a solution to an n × n inhomogeneous system Ax = b if A is a triangular matrix. Solution. False. (ii) Gaussian equivalent matrices have the same row space. Solution. True. ...
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This note was uploaded on 12/21/2011 for the course MAS 4301 taught by Professor Keithcornell during the Fall '11 term at UNF.

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