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Linear Algebra: Concepts and Methods Solutions to Problems Martin Anthony and Michele Harvey Department of Mathematics The London School of Economics and Political Science
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Linear Algebra: Concepts and Methods Solutions to Problems Martin Anthony and Michele Harvey Department of Mathematics The London School of Economics and Political Science2Introduction This docum... ent contains solutions to all the Problems from the book ‘Linear Algebra: Concepts and Methods’. It is intended as an aid to those who use the text in support of teaching a course in linear algebra. 34Chapter 1 Problem 1.1 (a) Ab is not defined. (b) Since C is a 3 × 3 matrix and A is 3 × 2, the product CA is defined and is the 3 × 2 matrix CA = 1 2 1 3 0 4 1 1 −1 2 1 1 1 0 3 = 4 6 6 0 9 8 (c) A + Cb is not defined, since Cb is 3 × 1 and A is 3 × 2. (d) A + D = 2 2 3 6 6 6 (e) bTD = ( 1 1 −1 ) 0 1 2 5 6 3 = ( −4 3 ) (f) The product DAT is a 3 × 3 matrix, as is C, so this expression is defined: DAT = 0 1 2 5 6 3 ( 2 1 0 1 1 3 ) = 15 9 9 1 1 3 9 7 15 , 56 CHAPTER 1. DAT + C = 15 9 9 1 1 3 9 7 15 + 1 2 1 3 0 4 1 1 −1 = 12 7 14 19 10 10 2 3 4 (g) bTb = ( 1 1 −1 ) −1 11 = (3). As mentioned in Section 1.8.2, we normally just identify a 1 × 1 matrix by its entry. So we may write (3) as 3. (h) bbT = −1 11 ( 1 1 −1 ) = −1 1 1 1 1 −1 1 − −1 1 (i) Cb = 1 2 1 3 0 4 1 1 −1 −1 11 = 2 4 4 Problem 1.2 The matrix aTb is the product of a 1 × n matrix and an n × 1 matrix, so it is a 1 × 1 matrix (which can be identified with the scalar it represents; that is, the scalar given by the inner product ⟨a, b⟩ ). The product bTa is also a 1 × 1 matrix. Since the product aTb is a 1 × 1 matrix, it is symmetric, so that aTb = (aTb)T = bTa. Problem 1.3 We have [Show More]
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