Answered: Let T(7) = Az for the following matrix… | bartleby

Algebra & Trigonometry with Analytic Geometry

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter9: Systems Of Equations And Inequalities

Section9.8: Determinants

Problem 6E

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What is the domain and codomain of T

O Let T() = Aã for the following matrix A and let 6 be the vector given below.
1
1
2 -4
А
0 -1
8.
1
3
1

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Define T:R2R2 by T(v)=projuv Where u is a fixed vector in R2. Show that the eigenvalues of A the standard matrix of T are 0 and 1.
Let A be an m×n matrix. Explain why the followingare true. Any vector b ∈ Rm can be uniquely written as a sumu + v, where u ∈ N(AT) and v ∈ R(A).
Let x and y be linearly independent vectors in Rnand let S = Span(x, y).We can use x and y to definea matrix A by settingA = xyT + yxT Show that N(A) = S⊥.
Let x and y be distinct vectors in Rn (i.e., x = y), and let A be an n × n matrix with the property that Ax = Ay. Show that det(A) = 0.
Let A be an m×n matrix. Explain why the followingare true. Any vector x in Rn can be uniquely written as a sumy + z, where y ∈ N(A) and z ∈ R(AT ).
Let Bj denote the matrix obtained by replacing thejth column of the identity matrix with a vectorb = (b1, . . . , bn)T . Use Cramer’s rule to show thatbj = det(Bj) for j = 1, . . . , n
Generate a matrix A by settingA = floor(10 ∗ rand(6))and generate a vector b by settingb = floor(20 ∗ rand(6, 1)) − 10 The free variable determined by the echelon form should be x3. By examining the system corresponding to the matrix U, you should be able to determine the solution corresponding to x3 = 0. Enter this solution into MATLAB as a column vector w. To check that Aw = c, compute the residual vector c − Aw.
Let A be an m × n matrix. Use the steps below to show that a vector x in Rn satisfies Ax=0 if and only if ATAx =0. This will show that Nul A = Nul ATA. a. Show that if Ax =0, then ATAx =0 b. Suppose ATAx = 0. Explain why xTATAx =0, and use this to show that Ax =0.
Gertrude owns a movie theatre in the town of Northern Millstone. Assume the following to be true. If a resident of Northern Millstone patronizes Gertrude’s movie theatre on any given day, the probability that this resident will patronize Gertrude’s movie theatre the next day is .05. If a resident of Northern Millstone does not patronize Gertrude’smovie theatre on any given day, the probability that this resident will not patronize Gertrude’s movie theatre the next day is .9. Find the equilibrium vector for the Markov chain described in this problem. (An equilibrium vector is also known as a stationary matrix.) Express the elements of the equilibrium vector as fractions.
For the matrix A below, find a nonzero vector in Nul A and a nonzero vector in Col A.
Find the coordinate matrix of x in Rn relative to the basis B'. B' = {(4, 3, 3), (−11, 0, 11), (0, 9, 2)}, x = (−22, −9, 20)
Find the coordinate matrix of x in Rn relative to the basis B′.B′ = {(4, 3, 3), (−11, 0, 11), (0, 9, 2)}, x = (11, 18, −7)

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