Differential Equations And Linear Algebra, Books A La Carte Edition (4th Edition)
4th Edition
ISBN: 9780321985811
Author: Stephen W. Goode, Scott A. Annin
Publisher: Pearson (edition 4)
expand_more
expand_more
format_list_bulleted
Concept explainers
Videos
Question
Chapter 5.1, Problem 26P
To determine
To sketch:
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solution
Students have asked these similar questions
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 5 Solutions
Differential Equations And Linear Algebra, Books A La Carte Edition (4th Edition)
Ch. 5.1 - True-False Review For Questions a-g, decide if the...Ch. 5.1 - True-False Review For Questions a-g, decide if the...Ch. 5.1 - True-False Review For Questions a-g, decide if the...Ch. 5.1 - True-False Review For Questions a-g, decide if the...Ch. 5.1 - Prob. 6TFRCh. 5.1 - Prob. 7TFRCh. 5.1 - Use the standard inner product in 5 to determine...Ch. 5.1 - Prob. 2PCh. 5.1 - If f(x)=sinx and g(x)=x on [0,], use the function...Ch. 5.1 - If f(x)=sinx and g(x)=2cosx+4 on [0,/2], use the...
Ch. 5.1 - Let m and n be positive real numbers. If f(x)=xm...Ch. 5.1 - If v=(2+i,32i,4+i) and w=(1+i,13i,3i), use the...Ch. 5.1 - If v=(63i,4,2+5i,3i) and w=(i,2i,3i,4i), use the...Ch. 5.1 - Let A=[a11a12a21a22] and B=[b11b12b21b22] be...Ch. 5.1 - Referring to A and B in Problem 8, show that the...Ch. 5.1 - Referring to A and B in Problem 8, show that the...Ch. 5.1 - Referring to A and B in Problem 8, show that the...Ch. 5.1 - For Problems 1213, use the inner product (5.1.13)...Ch. 5.1 - For Problems 1213, use the inner product (5.1.13)...Ch. 5.1 - Let p1(x)=a+bx and p2(x)=c+dx be vectors in P1(R)....Ch. 5.1 - Let V=C0[0,1] and for f and g in V, consider the...Ch. 5.1 - Let V=C0[0,1] and for f and g in V, consider the...Ch. 5.1 - Let V=C0[1,0] and for f and g in V, consider the...Ch. 5.1 - Consider the vector space R2. Define the mapping ,...Ch. 5.1 - For Problems 1921, determine the inner product of...Ch. 5.1 - For Problems 1921, determine the inner product of...Ch. 5.1 - For Problems 1921, determine the inner product of...Ch. 5.1 - Prob. 22PCh. 5.1 - Prob. 23PCh. 5.1 - Prob. 24PCh. 5.1 - Prob. 25PCh. 5.1 - Prob. 26PCh. 5.1 - Prob. 27PCh. 5.1 - Prob. 28PCh. 5.1 - Prob. 29PCh. 5.1 - Prob. 30PCh. 5.1 - Prob. 31PCh. 5.1 - Prob. 32PCh. 5.1 - Prob. 33PCh. 5.1 - Prob. 34PCh. 5.1 - Prob. 35PCh. 5.1 - Prob. 36PCh. 5.1 - Prob. 37PCh. 5.2 - Problems For Problems 1-5, determine whether the...Ch. 5.2 - Problems For Problems 1-5, determine whether the...Ch. 5.2 - Problems For Problems 1-5, determine whether the...Ch. 5.2 - Problems For Problems 1-5, determine whether the...Ch. 5.2 - Problems For Problems 1-5, determine whether the...Ch. 5.2 - Problems Let v=(7,2). Determine all non zero...Ch. 5.2 - Problems Let v=(3,6,1). Determine all vectors w in...Ch. 5.2 - Problems Let v1=(1,2,3), v2=(1,1,1). Determine all...Ch. 5.2 - Let v1=(4,0,0,1), v2=(1,2,3,4). Determine all...Ch. 5.2 - Problems For Problems 10-12, show that the given...Ch. 5.2 - Problems For Problems 10-12, show that the given...Ch. 5.2 - Prob. 12PCh. 5.2 - Prob. 13PCh. 5.2 - Prob. 14PCh. 5.2 - Prob. 15PCh. 5.2 - Prob. 16PCh. 5.2 - Prob. 17PCh. 5.2 - Prob. 18PCh. 5.2 - Prob. 19PCh. 5.2 - Prob. 20PCh. 5.2 - Prob. 21PCh. 5.2 - Prob. 22PCh. 5.2 - Prob. 23PCh. 5.2 - Prob. 24PCh. 5.2 - Problems For Problems 22-27, find the distance...Ch. 5.2 - Prob. 26PCh. 5.2 - For Problems 2227, find the distance from the...Ch. 5.2 - Problems For Problems 29-32, use result of problem...Ch. 5.2 - Problems For Problems 29-32, use result of problem...Ch. 5.2 - Prob. 31PCh. 5.2 - Problems For Problems 29-32, use result of problem...Ch. 5.2 - Problems Let {u1,u2,u3} be linearly independent...Ch. 5.2 - Prob. 34PCh. 5.3 - Problems For Problems 110, use the Gram-Schmidt...Ch. 5.3 - Problems For Problems 110, use the Gram-Schmidt...Ch. 5.3 - Problems For Problems 110, use the Gram-Schmidt...Ch. 5.3 - Prob. 4PCh. 5.3 - Prob. 5PCh. 5.3 - Prob. 6PCh. 5.3 - Prob. 7PCh. 5.3 - Prob. 8PCh. 5.3 - Prob. 9PCh. 5.3 - Prob. 10PCh. 5.3 - Prob. 15PCh. 5.3 - Prob. 16PCh. 5.3 - For problems 17-20, determine an orthogonal basis...Ch. 5.3 - For problems 17-20, determine an orthogonal basis...Ch. 5.3 - Prob. 25PCh. 5.4 - True-False Review For Questions a-f, decide if the...Ch. 5.4 - True-False Review For Questions a-f, decide if the...Ch. 5.4 - Prob. 3TFRCh. 5.4 - Prob. 4TFRCh. 5.4 - Prob. 5TFRCh. 5.4 - Prob. 6TFRCh. 5.4 - Prob. 1PCh. 5.4 - Prob. 2PCh. 5.4 - For problems 1-7, find the equation of the least...Ch. 5.4 - Prob. 4PCh. 5.4 - Prob. 5PCh. 5.4 - Prob. 6PCh. 5.4 - Prob. 7PCh. 5.4 - Prob. 8PCh. 5.4 - For Problems 8-9, find the equation of the least...Ch. 5.4 - Prob. 10PCh. 5.4 - Prob. 11PCh. 5.4 - Prob. 12PCh. 5.4 - Prob. 13PCh. 5.4 - Prob. 14PCh. 5.4 - If the size P(t) of a culture of bacteria measured...Ch. 5.4 - Prob. 16PCh. 5.4 - Prob. 17PCh. 5.4 - Prob. 18PCh. 5.5 - For Problem 1-2, determine the angle between the...Ch. 5.5 - Prob. 2APCh. 5.5 - Prob. 3APCh. 5.5 - Prob. 4APCh. 5.5 - Prob. 6APCh. 5.5 - For Problems 69, find an orthogonal basis for the...Ch. 5.5 - For Problems 69, find an orthogonal basis for the...Ch. 5.5 - For Problems 69, find an orthogonal basis for the...Ch. 5.5 - Prob. 11APCh. 5.5 - Prob. 12APCh. 5.5 - Prob. 13APCh. 5.5 - Prob. 14APCh. 5.5 - Prob. 15APCh. 5.5 - Prob. 16APCh. 5.5 - Prob. 17APCh. 5.5 - Prob. 18AP
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.Similar questions
- Question 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_forwardQuestion 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_forward
- Question 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_forwardSimplify the below expression. 3 - (-7)arrow_forward
- (6) ≤ a) Determine the following groups: Homz(Q, Z), Homz(Q, Q), Homz(Q/Z, Z) for n E N. Homz(Z/nZ, Q) b) Show for ME MR: HomR (R, M) = M.arrow_forward1. If f(x² + 1) = x + 5x² + 3, what is f(x² - 1)?arrow_forward2. What is the total length of the shortest path that goes from (0,4) to a point on the x-axis, then to a point on the line y = 6, then to (18.4)?arrow_forward
- موضوع الدرس Prove that Determine the following groups Homz(QZ) Hom = (Q13,Z) Homz(Q), Hom/z/nZ, Qt for neN- (2) Every factor group of adivisible group is divisble. • If R is a Skew ficald (aring with identity and each non Zero element is invertible then every R-module is free.arrow_forwardI have ai answers but incorrectarrow_forwardwhat is the slope of the linear equation-5x+2y-10=0arrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage
Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:9781133382119
Author:Swokowski
Publisher:Cengage
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:9781305658004
Author:Ron Larson
Publisher:Cengage Learning
Vector Spaces | Definition & Examples; Author: Dr. Trefor Bazett;https://www.youtube.com/watch?v=72GtkP6nP_A;License: Standard YouTube License, CC-BY
Understanding Vector Spaces; Author: Professor Dave Explains;https://www.youtube.com/watch?v=EP2ghkO0lSk;License: Standard YouTube License, CC-BY