EBK ALGEBRA FOUNDATIONS
15th Edition
ISBN: 9780321978929
Author: Martin-Gay
Publisher: PEARSON
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Chapter F, Problem 10ES
To determine
To solve:the givenset of linearequationfor value of variable using Cramer’s rule of matrices and mathematical methodologies.
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Într-un bloc sunt apartamente cu 2 camere și apartamente cu 3 camere , în total 20 de apartamente și 45 de camere.Calculați câte apartamente sunt cu 2 camere și câte apartamente sunt cu 3 camere.
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]
Chapter F Solutions
EBK ALGEBRA FOUNDATIONS
Ch. F - Evaluate. See Example 1. 3517Ch. F - Prob. 2ESCh. F - Prob. 3ESCh. F - Prob. 4ESCh. F - Prob. 5ESCh. F - Prob. 6ESCh. F - Prob. 7ESCh. F - Prob. 8ESCh. F - Prob. 9ESCh. F - Prob. 10ES
Ch. F - Prob. 11ESCh. F - Prob. 12ESCh. F - Prob. 13ESCh. F - Prob. 14ESCh. F - Prob. 15ESCh. F - Prob. 16ESCh. F - Prob. 17ESCh. F - Prob. 18ESCh. F - Prob. 19ESCh. F - Prob. 20ESCh. F - Prob. 21ESCh. F - Prob. 22ESCh. F - Prob. 23ESCh. F - Prob. 24ESCh. F - Prob. 25ESCh. F - Prob. 26ESCh. F - Prob. 27ESCh. F - Prob. 28ESCh. F - Prob. 29ESCh. F - Prob. 30ESCh. F - Prob. 31ESCh. F - Prob. 32ESCh. F - Prob. 33ESCh. F - Prob. 34ESCh. F - Prob. 35ESCh. F - Prob. 36ESCh. F - Prob. 37ESCh. F - Prob. 38ESCh. F - Prob. 39ESCh. F - Prob. 40ESCh. F - Prob. 41ESCh. F - Prob. 42ESCh. F - Prob. 43ESCh. F - Prob. 44ESCh. F - Prob. 45ESCh. F - Prob. 46ESCh. F - Prob. 47ESCh. F - Prob. 48ESCh. F - Prob. 49ESCh. F - Prob. 50ESCh. F - Prob. 51ESCh. F - Prob. 52ESCh. F - Find the value of each determinant. To evaluate a...Ch. F - Prob. 54ESCh. F - Find the value of each determinant. To evaluate a...Ch. F - Find the value of each determinant. To evaluate a...
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- 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]arrow_forwardQuestion 4 (a) The following matrices represent linear maps on R² with respect to an orthonormal basis: = [1/√5 2/√5 [2/√5 -1/√5] " [1/√5 2/√5] A = B = [2/√5 1/√5] 1 C = D = = = [ 1/3/5 2/35] 1/√5 2/√5 -2/√5 1/√5' For each of the matrices A, B, C, D, state whether it represents a self-adjoint linear map, an orthogonal linear map, both, or neither. (b) For the quadratic form q(x, y, z) = y² + 2xy +2yz over R, write down a linear change of variables to u, v, w such that q in these terms is in canonical form for Sylvester's Law of Inertia. [6] [4]arrow_forwardpart b pleasearrow_forward
- Question 5 (a) Let a, b, c, d, e, ƒ Є K where K is a field. Suppose that the determinant of the matrix a cl |df equals 3 and the determinant of determinant of the matrix a+3b cl d+3e f ГЪ e [ c ] equals 2. Compute the [5] (b) Calculate the adjugate Adj (A) of the 2 × 2 matrix [1 2 A = over R. (c) Working over the field F3 with 3 elements, use row and column operations to put the matrix [6] 0123] A = 3210 into canonical form for equivalence and write down the canonical form. What is the rank of A as a matrix over F3? 4arrow_forwardQuestion 2 In this question, V = Q4 and - U = {(x, y, z, w) EV | x+y2w+ z = 0}, W = {(x, y, z, w) € V | x − 2y + w − z = 0}, Z = {(x, y, z, w) € V | xyzw = 0}. (a) Determine which of U, W, Z are subspaces of V. Justify your answers. (b) Show that UW is a subspace of V and determine its dimension. (c) Is VU+W? Is V = UW? Justify your answers. [10] [7] '00'arrow_forwardTools Sign in Different masses and Indicated velocities Rotational inert > C C Chegg 39. The balls shown have different masses and speeds. Rank the following from greatest to least: 2.0 m/s 8.5 m/s 9.0 m/s 12.0 m/s 1.0 kg A 1.2 kg B 0.8 kg C 5.0 kg D C a. The momenta b. The impulses needed to stop the balls Solved 39. The balls shown have different masses and speeds. | Chegg.com Images may be subject to copyright. Learn More Share H Save Visit > quizlet.com%2FBoyE3qwOAUqXvw95Fgh5Rw.jpg&imgrefurl=https%3A%2F%2Fquizlet.com%2F529359992%2Fc. Xarrow_forward
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