FlipIt for College Physics (Algebra Version - Six Months Access)
17th Edition
ISBN: 9781319032432
Author: Todd Ruskell
Publisher: W.H. Freeman & Co
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Question
Chapter 11, Problem 93QAP
To determine
The ratio of
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Check out a sample textbook solutionStudents have asked these similar questions
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]
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]
Chapter 11 Solutions
FlipIt for College Physics (Algebra Version - Six Months Access)
Ch. 11 - Prob. 1QAPCh. 11 - Prob. 2QAPCh. 11 - Prob. 3QAPCh. 11 - Prob. 4QAPCh. 11 - Prob. 5QAPCh. 11 - Prob. 6QAPCh. 11 - Prob. 7QAPCh. 11 - Prob. 8QAPCh. 11 - Prob. 9QAPCh. 11 - Prob. 10QAP
Ch. 11 - Prob. 11QAPCh. 11 - Prob. 12QAPCh. 11 - Prob. 13QAPCh. 11 - Prob. 14QAPCh. 11 - Prob. 15QAPCh. 11 - Prob. 16QAPCh. 11 - Prob. 17QAPCh. 11 - Prob. 18QAPCh. 11 - Prob. 19QAPCh. 11 - Prob. 20QAPCh. 11 - Prob. 21QAPCh. 11 - Prob. 22QAPCh. 11 - Prob. 23QAPCh. 11 - Prob. 24QAPCh. 11 - Prob. 25QAPCh. 11 - Prob. 26QAPCh. 11 - Prob. 27QAPCh. 11 - Prob. 28QAPCh. 11 - Prob. 29QAPCh. 11 - Prob. 30QAPCh. 11 - Prob. 31QAPCh. 11 - Prob. 32QAPCh. 11 - Prob. 33QAPCh. 11 - Prob. 34QAPCh. 11 - Prob. 35QAPCh. 11 - Prob. 36QAPCh. 11 - Prob. 37QAPCh. 11 - Prob. 38QAPCh. 11 - Prob. 39QAPCh. 11 - Prob. 40QAPCh. 11 - Prob. 41QAPCh. 11 - Prob. 42QAPCh. 11 - Prob. 43QAPCh. 11 - Prob. 44QAPCh. 11 - Prob. 45QAPCh. 11 - Prob. 46QAPCh. 11 - Prob. 47QAPCh. 11 - Prob. 48QAPCh. 11 - Prob. 49QAPCh. 11 - Prob. 50QAPCh. 11 - Prob. 51QAPCh. 11 - Prob. 52QAPCh. 11 - Prob. 53QAPCh. 11 - Prob. 54QAPCh. 11 - Prob. 55QAPCh. 11 - Prob. 56QAPCh. 11 - Prob. 57QAPCh. 11 - Prob. 58QAPCh. 11 - Prob. 59QAPCh. 11 - Prob. 60QAPCh. 11 - Prob. 61QAPCh. 11 - Prob. 62QAPCh. 11 - Prob. 63QAPCh. 11 - Prob. 64QAPCh. 11 - Prob. 65QAPCh. 11 - Prob. 66QAPCh. 11 - Prob. 67QAPCh. 11 - Prob. 68QAPCh. 11 - Prob. 69QAPCh. 11 - Prob. 70QAPCh. 11 - Prob. 71QAPCh. 11 - Prob. 72QAPCh. 11 - Prob. 73QAPCh. 11 - Prob. 74QAPCh. 11 - Prob. 75QAPCh. 11 - Prob. 76QAPCh. 11 - Prob. 77QAPCh. 11 - Prob. 78QAPCh. 11 - Prob. 79QAPCh. 11 - Prob. 80QAPCh. 11 - Prob. 81QAPCh. 11 - Prob. 82QAPCh. 11 - Prob. 83QAPCh. 11 - Prob. 84QAPCh. 11 - Prob. 85QAPCh. 11 - Prob. 86QAPCh. 11 - Prob. 87QAPCh. 11 - Prob. 88QAPCh. 11 - Prob. 89QAPCh. 11 - Prob. 90QAPCh. 11 - Prob. 91QAPCh. 11 - Prob. 92QAPCh. 11 - Prob. 93QAPCh. 11 - Prob. 94QAP
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- part 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_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_forward
- Tools 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_forward
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