New PDF release: A short linear algebra book (answers)

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15 The solution set is  4x2 − 3x3 + x4   x2  x2 , x3 , x4 ∈ R} {   x3 x4  so a natural basis is this       4 −3 1 1  0  0  , ,  0  1  0 0 0 1 (checking linear independence is easy). Thus the dimension is three. 17 (a) As in the prior exercise, the space M2×2 of matrices without restriction has this basis 1 0 0 0 , 0 0 1 0 0 0 0 , , 0 1 0 0 1 and so the dimension is four. (b) For this space { a b c d a = b − 2c and d ∈ R} = {b · 1 1 −2 0 0 0 +c· +d· 0 0 1 0 0 1 b, c, d ∈ R} this is a natural basis.

Answers for Topic: Fields 1 These checks are all routine; most consist only of remarking that property is so familiar that it does not need to be proved. 2 For both of these structures, these checks are all routine. As with the prior question, most of the checks consist only of remarking that property is so familiar that it does not need to be proved. 3 There is no multiplicative inverse for 2, so the integers do not satisfy condition (5). 4 These checks can be done by listing all of the possibilities.

26 Considering a function to be a set, specifically, a set of ordered pairs (x, f (x)), then the only function with an empty domain is the empty set. Thus this is a trivial vector space, and has dimension zero. 8. 28 The first chapter defines a plane — a ‘2-flat’ — to be a set of the form {p + t1 v1 + t2 v2 t1 , t2 ∈ R} (also there is a discussion of why this is equivalent to the description often taken in Calculus as the set of points (x, y, z) subject to some linear condition ax + by + cz = d).

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A short linear algebra book (answers) by Hefferon J.

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