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Linear Algebra A Modern Introduction 4th edition David Poole - Solutions

Prove that if A and B are square matrices and AB is invertible, then both A and B are invertible.
In Exercises, use the Gauss-Jordan method to find the inverse of the given matrix (if it exists).a.b. c.
Partitioning large square matrices can sometimes make their inverses easier to compute, particularly if the blocks have a nice form. In Exercises, verify by block multiplication that the inverse of a matrix, if partitioned as shown, is as claimed. (Assume that all inverses exist as needed.)a.b. c.
In Exercises, partition the given matrix so that you can apply one of the formulas from Exercises, and then calculate the inverse using that formula.a.b.
In Exercises, solve the system Ax = b using the given LU factorization of A.a.b.
Generalize the definition of LU factorization to non square matrices by simply requiring U to be a matrix in row echelon form. With this modification, find an LU factorization of the matrices in Exercises.a.b.
For a n invertible matrix with a n LU factorization A = LU, both L and U will be invertible and A-l = U-1 L-1. In Exercises, find L-1, U-1, and A-1 for the given matrix.a. A in Exercise 1b. A in Exercise 4
The inverse of a matrix can also be computed by solving several systems of equations using the method of Example 3.34. For an n X n matrix A, to find its inverse we need to solve AX = In for the n Ã— n matrix X. Writing this equation as A [x1 x2 · · xn] = [ e1 e2 ·
In Exercises, write the given permutation matrix as a product of elementary (row interchange) matricesa.b. c.
In Exercises, find a PTLU factorization of the given matrix A.a.b.
Prove that there are exactly n! n × n permutation matrices.
In Exercises, solve the system Ax = b using the given factorization A = PTLU. Because PPT = I, PT LUx = b can be rewritten as LUx = Pb. This system can then be solved using the method of Example 3.34.a.b.
Prove that a product of unit lower triangular matrices is unit lowers triangular.
Prove that every unit lower triangular matrix is invertible and that its inverse is also unit lowers triangular.
An LDU factorization of a square matrix A is a factorization A = LDU, where L is a unit lower triangular matrix, D is a diagonal matrix, and U is a unit upper triangular matrix (upper triangular with l s on its diagonal). In Exercises, find an LDU factorization of A.a. A in Exercise 1b. A in
If A is symmetric and invertible and has an LDU factorization, show that U = LT.
If A is symmetric and invertible and A = LDLT (with L unit lower triangular and D diagonal), prove that this factorization is unique. That is, prove that if we also have A = L1D1LT1 (with L1 unit lower triangular and D1 diagonal), then L = L1 and D = D1.
In Exercises, find an LU factorization of the given matrix.a.b. c.
In Exercises, let S be the collection of vectorsThat satisfies the given property. In each case, either proves that S forms a subspace of R2 or gives a counterexample to show that it does not. a. x = 0 b. x ‰¥ 0, y ‰¥ 0 c. y = 2x
In Exercise, determine whether b is in col (A) and whether w is in row(A), as in Example 3.4 1 .w = [-1 1 1]
In Exercise 11, determine whether w is in row (A), using the method described in the Remark following Example 3.4 1.In Exercise 11w = [-1 1 1]
In Exercise 12, determine whether w is in row(A), using the method described in the Remark following Example 3.4 1 .In Exercise 12w = [2 4 -5]
If A the matrix in Exercise 11 isin null (A)? In Exercise 11
If A is the matrix in Exercise 12, isin null(A)?
In Exercises, give bases for row (A), col (A), and null (A).a.b.
In Exercises, find bases for row (A) and col (A) in the given exercises using AT.a. Exercise 17b. Exercise 18
Explain carefully why your answers to Exercises 17 and 21 are both correct even though there appear to be differences.In Exercise 17
Explain carefully why your answers to Exercise 18 are both correct even though there appear to be differences.In Exercise 18
In Exercises, find a basis for the span of the given vectors.a.b.
For Exercises, find bases for the spans of the vectors in the given exercises from among the vectors themselves. a. In Exercise 29 [2 -3 1], [1 -1 0], [4 -4 1] b. In Exercise 30 [0 1 -2 1], [3 1 -1 0], [2 1 5 1]
Prove that if R is a matrix in echelon form, then a basis for row(R) consists of the nonzero rows of R.
For Exercises, give the rank and the nullity of the matrices in the given exercises.a. In Exercise 17b. In Exercise 18 c. In Exercise 19
If A is a 3 × 5 matrix, explain why the columns of A must be linearly dependent.
If A is a 4 × 2 matrix, explain why the rows of A must be linearly dependent.
If A is a 3 × 5 matrix, what are the possible values of nullity (A)?
If A is a 4 × 2 matrix, what are the possible values of nullity (A)?
In Exercises, find all possible values of rank (A) as a varies.a.b.
Exercises by considering the matrix with the given vectors as its columnsa. Doform a basis for R3? b. Do from the basis for R3?
In Exercise, let S be the collection of vectorsThat satisfies the given property. In each case, either prove that S forms a subspace of R3 or give a counterexample to show that it does not. a. x = y = z b. z = 2x, y = 0
In Exercises, show that w is in span (B) and find the coordinate vector [w]B.a.b.
In Exercises, compute the rank and nullity of the given matrices over the indicated ZPa.Over Z2 b. Over Z3 c. Over Z5
If A is m × n, prove that every vector in null (A) is orthogonal to every vector in row (A).
If A and B are n × n matrices of rank n, prove that AB has rank n.
(a) Prove that rank(AB) ≤ rank(B). (b) Give an example in which rank (AB) < rank (B).
(a) Prove that rank(AB) ≤ rank (A). (b) Give an example in which rank (AB) < rank (A).
(a) Prove that if U is invertible, then rank (UA) = rank (A). (b) Prove that if V is invertible, then rank (AV) = rank (A).
Prove that an m × n matrix A has rank 1 if and only if A can be written as the outer product uvT of a vector u in Rm and v in Rn.
If an m × n matrix A has rank r, prove that A can be written as the sum of r matrices, each of which has rank 1.
Prove that, for m × n matrices A and B, rank (A + B) ≤ rank (A) + rank (B).
Let A be an n × n matrix such that A2 = O. Prove that rank (A) . . . n/2.
Let A be a skew - symmetric n × n matrix. (a) Prove that xT Ax = 0 for all x in Rn. (b) Prove that I + A is invertible.
Prove that every line through the origin in R3 is a subspace of R3.
Let TA : R2 †’ R2 be the matrix transformation corresponding toFind TA (u) and TA (v), Where And
In Exercises, find the standard matrix of the linear transformation in the given exercise.a.b.
In Exercise, show that the given transformation from R2 to R2 is linear by showing that it is a matrix transformation a. F reflects a vector in the y-axis. b. R rotates a vector 45° counterclockwise about the origin. c. D stretches a vector by a factor of 2 in the x-component and a factor of 3 in
The three types of elementary matrices give rise to five types of 2 Ã— 2 matrices with one of the following forms:Each of these elementary matrices corresponds to a linear transformation from R2 to R2. Draw pictures to illustrate the effect of each one on the unit square with vertices
Let TA : R2 †’ R3 be the matrix transformation corresponding toFind TA (u) and Where And
In Exercise, find the standard matrix of the given linear transformation from R2 o R2. a. Counterclockwise rotation through 120o about the origin b. Clockwise rotation through 30o about origin c. Projection onto the line y = 2x
Let e be a line through the origin in R2, Pl the linear transformation that projects a vector onto l, and Fl the transformation that reflects a vector in l.(a) Draw diagrams to show that Fl is linear.(b) Figure 3.14 suggests a way to find the matrix of Fe, using the fact that the diagonals of a
In Exercises apply part (b) or (c) of Exercise 26 to find the standard matrix of the transformation.In Exercise 26(b) Figure 3.14 suggests a way to find the matrix of Fe, using the fact that the diagonals of a parallelogram bisect each other. Prove that Fl(x) = 2Pl(x) - x, and use this result to
Check the formula for S o T in Example 3.60, by performing the suggested direct substitution.
In Exercises, prove that the given transformation is a linear transformation, using the definition.a.b.
In Exercise, verify Theorem 3.32 by finding the matrix of S o T (a) by direct substitution and (b) by matrix multiplication of [S] [T].a.b.
In Exercises, find the standard matrix of the composite transformation from R2 to R2. a. Counterclockwise rotation through 60°, followed by reflection in the line y = x b. Reflection in the y-axis, followed by clockwise rotation through 30°
In Exercises, use matrices to prove the given statements about transformations from R2 to R2 a. If Rθ denotes a rotation (about the origin) through the angle θ, then Rα o Rβ = Rα+β b. If θ is the angle between lines e and m (through the origin), then Fm ° Fl = R+2θ·
Let T be a linear transformation from R2 to R2 (or from R3 to R3). Prove that T maps a straight line to a straight line or a point.
Let T be a linear transformation from R2 to R2 (or from R3 to R3). Prove that T maps parallel lines to parallel lines, a single line, a pair of points, or a single point.
In Exercises, let ABCD be the square with vertices (- 1, 1), (1, 1), (1, - 1), and (- l, - 1). Use the result in Exercises to find and draw the image of ABCD under the given transformation.a. T in Exercise 3In Exercise 17, 18 show that the given transformation from R2 to R2 is linear by showing
Prove that Pl (cv) = cPl (v) for any scalar c
Prove that T: Rn → Rm is a linear transformation if and only if T (c1v1 + c2v2) = c1T(v1) + c2T (v2) For all v1, v2 in Rn and scalars c1, c2
Prove that (as noted at the beginning of this section) the range of a linear transformation T : Rn → Rm is the column space of its matrix [T].
If A is an invertible 2 × 2 matrix, what does the fundamental Theorem of Invertible Matrices assert about the corresponding linear transformation TA in light of Exercise 19?
In Exercise, give a counterexample to show that the given transformation is not a linear transformation.a.b. c.
Data have been accumulated on the heights of children relative to their parents. Suppose that the probabilities that a tall parent will have a tall, medium - height, or short child are 0.6, 0.2 and 0.2, respectively; the probabilities that a medium height parent will have a tall medium height or
A study of pinon (pine) nut crops in the American southwest from 1940 to 1947 hypothesized that nut production followed a Markov chain. The data suggested that if one year's crop was good, then the probabilities that the following year's crop would be good, fair, or poor were 0.08, 0.07, and 0.85
Robots have been programmed to traverse the maze shown in Figure and at each junction randomly choose which way to go.a. construct the transition matrix for the markov chain that models this situation. b. Suppose we start with 15 robots at each junction. Find the steady state distribution of
Let j denote a row vector consisting entirely of 1s. Prove that a nonnegative matrix P is a stochastic matrix if and only if j P = j.
a. Show that the product of two 2 × 2 stochastic matrices is also a stochastic matrix. b. Prove that the product of two n × n stochastic matrices is also a stochastic matrix c. If a 2 × 2 stochastic matrix p is invertible, prove that p-1 is also a stochastic matrix.
In Exercise 9, if Monday is a dry day, what is the expected number of days until a wet day? Suppose we want to know the average (or expected) number of steps it will take to go from state i to state j in a Markov chain. It can be shown that the following computation answers this question: Delete
In Exercise 10, what is the expected number of generations until a short person has a tall descendant? Suppose we want to know the average (or expected) number of steps it will take to go from state i to state j in a Markov chain. It can be shown that the following computation answers this
In Exercise 11, if the pinon nut crop is fair one year, what is the expected number of years until a good crop occurs? Suppose we want to know the average (or expected) number of steps it will take to go from state i to state j in a Markov chain. It can be shown that the following computation
In Exercise 12, starting from each of the other junctions, what is the expected number of moves until a robot reaches junction 4?Suppose we want to know the average (or expected) number of steps it will take to go from state i to state j in a Markov chain. It can be shown that the following
In Exercises, determine which of the matrices are exchange matrices. For those that are exchange matrices, find a nonnegative price vector that satisfies Equation (1).a.b. c.
In Exercises, determine whether the given consumption matrix is productive.a.b. c.
In Exercises, a consumption matrix C and a demand vector d are given. In each case, find a feasible production vector x that satisfies Equation (2).a.b.
Let A, B, C, and D be n × n matrices and x and y vectors in Rn. Prove following inequalities; a. If A ≥ B ≥ O and C ≥ D ≥ O, then AC ≥ BD ≥ O. b. If A > B and x ≥ 0, x ≠ 0, then Ax > Bx.
A population with two age classes has a Leslie matrixIf the initial population vector is Compute x1, x2, and x3.
A population with three age classes has a Leslie matrixIf the initial population vector is Compute x1, x2, and x3.
A population with three age classes has a Leslie matrixIf the initial population vector is Compute x1, x2 and x3.
A population with four age classes has a Leslie matrixIf the initial population vector is Compute x1, x2, and x3.
A certain species with two age classes of 1 year's duration has a survival probability of 80% from class 1 to class 2. Empirical evidence shows that, on average, each female gives birth to five females per year. Thus two possible Leslie matrices areAnd a. Starting with Compute x1 ....x10 in each
Suppose the Leslie matrix for the VW beetle is L =Starting with an arbitrary x0 determine the behavior of this population.
Suppose the Leslie matrix for the VW beetle isInvestigate the effect of varying the survival probability s of the young beetles.
Woodland caribou are found primarily in the western provinces of Canada and the American northwest. The average lifespan of a female is about 14 years. The birth and survival rate for each age bracket are given in Table 3.4, which shows that caribou cows do not give birth at all during their first
In Exercises, determine the adjacency matrix of the given graph.a.b. c.
In Exercises, draw a graph that has the given adjacency matrix.a.b. c.
In Exercises, determine the adjacency matrix of the given digraph.a.b. c.
In Exercises, draw a digraph that has the given adjacency matrix.a.b. c.
In Exercises, use powers of adjacency matrices to determine the number of paths of the specified length between the given vertices.a. Exercise 50, Length 2, v1 and v2.b. Exercise 52, Length 2, v1 and v2. c. Exercise 50, Length 3, v1 and v3.
Let A be the adjacency matrix of a graph G. a. If row I of A is all zeros, what does this imply about G? b. If column j of A is all zeros, what does this imply about G?

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