15th Edition

ISBN: 9781938168994

Author: OpenStax

Publisher: OpenStax

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Textbook Question

Chapter 2.2, Problem 80E

In the following exercises, evaluate the expression for the given value.

x 2 + 5 x − 8 when x = 6

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Chapter 2.2, Problem 79E

Chapter 2.2, Problem 81E

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1.2.19. Let and s be natural numbers. Let G be the simple graph with vertex set Vo... V„−1 such that v; ↔ v; if and only if |ji| Є (r,s). Prove that S has exactly k components, where k is the greatest common divisor of {n, r,s}.

Question 3 over a field K. In this question, MË(K) denotes the set of n × n matrices (a) Suppose that A Є Mn(K) is an invertible matrix. Is it always true that A is equivalent to A-¹? Justify your answer. (b) Let B be given by 8 B = 0 7 7 0 -7 7 Working over the field F2 with 2 elements, compute the rank of B as an element of M2(F2). (c) Let 1 C -1 1 [4] [6] and consider C as an element of M3(Q). Determine the minimal polynomial mc(x) and hence, or otherwise, show that C can not be diagonalised. [7] (d) Show that C in (c) considered as an element of M3(R) can be diagonalised. Write down all the eigenvalues. Show your working. [8]

R denotes the field of real numbers, Q denotes the field of rationals, and Fp denotes the field of p elements given by integers modulo p. You may refer to general results from lectures. Question 1 For each non-negative integer m, let R[x]m denote the vector space consisting of the polynomials in x with coefficients in R and of degree ≤ m. x²+2, V3 = 5. Prove that (V1, V2, V3) is a linearly independent (a) Let vi = x, V2 = list in R[x] 3. (b) Let V1, V2, V3 be as defined in (a). Find a vector v € R[×]3 such that (V1, V2, V3, V4) is a basis of R[x] 3. [8] [6] (c) Prove that the map ƒ from R[x] 2 to R[x]3 given by f(p(x)) = xp(x) — xp(0) is a linear map. [6] (d) Write down the matrix for the map ƒ defined in (c) with respect to the basis (2,2x + 1, x²) of R[x] 2 and the basis (1, x, x², x³) of R[x] 3. [5]

Chapter 2 Solutions

PREALGEBRA

Ch. 2.1 - Translate from algebra to words. 18+11 (b) (27)(9)...Ch. 2.1 - Translate from algebra to words. 4719 (b) 729 (c)...Ch. 2.1 - Translate from algebra to words. 1427 (b) 1928 (c)...Ch. 2.1 - Translate from algebra to words. 1915 (b) 7=125...Ch. 2.1 - Use Figure 2.2 to fill in the appropriate symbol....Ch. 2.1 - Use Figure 2.2 to fill in the appropriate symbol....Ch. 2.1 - Determine if each is an expression or an equation:...Ch. 2.1 - Determine if each is an expression or an equation:...Ch. 2.1 - Write each expression in exponential form:...Ch. 2.1 - Write each expression in exponential form:...

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