136A5 - L . 3. [5 marks] A hyperplane in R n is a subspace...

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ASSIGNMENT 5 for SECTION 001 Questions from the textbook Section 3-4: B3 (a), (d) and (e) [3 marks] ; B4 [2 marks] ; B6 [2 marks] ; B10 [3 marks] and D3 [3 marks] Section 4-1: B1 (a), (d) and (e) [3 marks] ; B2 (a) and (b) [2 marks] ; and D4 [6 marks] Section 4-2: B5 [3 marks] , D1 [2 mark] , D3 [3 marks] , D4 [3 marks] and D7 [3 marks] In addition to the questions listed here, do the MATLAB question from Assignment 5 for the other sections; this may be found under Content > Assignments > Assignment 5 [2 marks] Other questions 1. [5 marks] Let x 1 ,..., x m and y 1 ,..., y m be vectors in R n such that a 11 x 1 + a 12 x 2 + ··· + a 1 m x m = y 1 a 21 x 1 + a 22 x 2 + ··· + a 2 m x m = y 2 . . . a m 1 x 1 + a m 2 x 2 + ··· + a mm x m = y m and Rank a 11 a 12 ··· a 1 m a 21 a 22 ··· a 2 m . . . . . . . . . . . . a m 1 a m 2 ··· a mm = m. Prove that Span { x 1 ,..., x m } = Span { y 1 ,..., y m } . 2. [5 marks] Let L : P 3 R 3 be the linear transformation given by L : f ( x ) 7→ f (1) f (2) f (3) . Find the range and kernel of
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Unformatted text preview: L . 3. [5 marks] A hyperplane in R n is a subspace of the form x 1 . . . x n : a 1 x 1 + + a n x n = 0 , where the scalars a 1 ,...,a n are not all zero. Prove that such hyperplanes are of dimension n-1, and that every subspace in R n of dimension n-1 is equal to a hyperplane. 4. [5 marks] Let V be a vector space, and U 1 ,U 2 V be subspaces. Prove that dim( U 1 + U 2 ) = dim U 1 + dim U 2-dim( U 1 U 2 ) . (You need not prove that U 1 + U 2 and U 1 U 2 are subspaces; your proof from question D3 of Section 3-4 is easily generalized.)...
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