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linear algebra and its applications

Linear Algebra And Its Applications 6th Global Edition David Lay, Steven Lay, Judi McDonald - Solutions

In R4, let v1 = (1,-1,2,-1), v2(1, 0, 2,0), and v3 (1, 0, 2, 0), and v4 = (1, 0, 3, 1). a. Show that set {v1,v2,v3,v4} is affinely independent.b. Let A = aff {v1, v2, v3, v4) and B = [f: 3], where f is the linear functional defined by f(x1, x2, x3, x4) = x1 + x2 +
Prove the given statement about subsets A and B of Rn, or provide the required example in R2. A proof for an exercise may use results from earlier exercises (as well as theorems already available in the text).Find an example in R2 to show that equality need not hold in the statement of Exercise
Prove the given statement about subsets A and B of Rn, or provide the required example in R2. A proof for an exercise may use results from earlier exercises (as well as theorems already available in the text).If A ⊆ B, then aff A ⊆ aff B.
Deal with the following concepts: A point p is called a positive combination of the points v1,........., vk if p = c1v1 +..........+ckvk, with all ci ≥ 0. The set of all positive combinations of points of a set S is called the positive hull of S and is denoted by pos S.Let S = {(–1,
Give an example of a closed subset S of R2 such that conv S is not closed.
Deal with the following concepts: A point p is called a positive combination of the points v1,........., vk if p = c1v1 +..........+ckvk, with all ci ≥ 0. The set of all positive combinations of points of a set S is called the positive hull of S and is denoted by pos S.What special
Prove the given statement about subsets A and B of Rn, or provide the required example in R2. A proof for an exercise may use results from earlier exercises (as well as theorems already available in the text).aff (A ∩ B) ⊆ (aff A ∩ aff B).
Deal with the following concepts: A point p is called a positive combination of the points v1,........., vk if p = c1v1 +..........+ckvk, with all ci ≥ 0. The set of all positive combinations of points of a set S is called the positive hull of S and is denoted by pos S.Let S be a
Let M be the matrix game having payoff matrix Find E(x, y), ν(x), and v(y) when x and y have the given values. 1 0 3 2-2 14 -1 1
Let M be the matrix game having payoff matrix Find E(x, y), ν(x), and v(y) when x and y have the given values. 201 -1 1-2 1-2 2 -1 0 1
Find all saddle points for the matrix games. 4 1 3]
Find all saddle points for the matrix games. 3 4 1-5 4 3 7 5 -2 3 2 3
Find all saddle points for the matrix games. -2 4 3 5 2 1 -1 2 2 1-3 -3 0 0
Find all saddle points for the matrix games. 2 1 4 -2 3 1
In write the payoff matrix for each game.Player R has a supply of dimes and quarters. Player R chooses one of the coins, and player C must guess which coin R has chosen. If the guess is correct, C takes the coin. If the guess is incorrect, C gives R an amount equal to R’s chosen coin.
In write the payoff matrix for each game. Players R and C each show one, two, or three fingers. If the total number N of fingers shown is even, then C pays N dollars to R. If N is odd, R pays N dollars to C.
In write the payoff matrix for each game. In the traditional Japanese children’s game janken (or “rock, scissors, paper”), at a given signal, each of two players shows either no fingers (rock), two fingers (scissors), or all five (paper). Rock beats scissors, scissors beats paper, and paper
Find the optimal row and column strategies and the value of each matrix game. [1 4 6 3 2 2 0 5
In write the payoff matrix for each game. Player R has three cards: a red 3, a red 6, and a black 7. Player C has two cards: a red 4 and a black 9. They each show one of their cards. If the cards are the same color, R receives the larger of the two numbers. If the cards are of different colors, C
Find the optimal row and column strategies and the value of each matrix game. [³ 3-2 1
Find the optimal row and column strategies and the value of each matrix game. 2 -2 6 -3
Find the optimal row and column strategies and the value of each matrix game. 3 5 4 1
Find the optimal row and column strategies and the value of each matrix game. 3 - 1 [- 5 3 918 2
Find the optimal row and column strategies and the value of each matrix game. 0 1 −1 2 -1 1 -1 4 3 −1 4 0-2 −1 3 -3 0-2 5 1 -
Find the optimal row and column strategies and the value of each matrix game. 5 4 -2 -1 2 -3 1 3 1
Find the optimal row and column strategies and the value of each matrix game. 6 0 6 2 4 5 4 2 3 5 5 3 5 7 2 7
Mark each statement True or False (T/F). Justify each answer.The payoff matrix for a matrix game indicates what R wins for each combination of moves.
Find the optimal strategies and the value of the game in Example 2.Data From Example 2 EXAMPLE 2 Again suppose that each player has a supply of pennies, nickels, and dimes to play, but this time the payoff matrix is given as follows: P Player R n d Column maxima P 10 1 Player C n -5 10 1 0 -10 1 d
A certain army is engaged in guerrilla warfare. It has two ways of getting supplies to its troops: it can send a convoy up the river road or it can send a convoy overland through the jungle. On a given day, the guerrillas can watch only one of the two roads. If the convoy goes along the river and
Suppose in Exercise 19 that whenever the convoy goes overland two soldiers are lost to land mines, whether they are attacked or not. Thus, if the army encounters the guerrillas, there will be 9 casualties. If it does not encounter the guerrillas, here will be 2 casualties.a. Find the optimal
Bill and Wayne are playing a game in which each player has a choice of two colors: red or blue. The payoff matrix with Bill as the row player is given below.For example, this means that if both people choose red, then Bill pays Wayne one unit.a. Using the same payoffs for Bill and Wayne, write the
Mark each statement True or False (T/F). Justify each answer.If aij is a saddle point, then aij is the smallest entry in row i and the largest entry in column j .
Mark each statement True or False (T/F). Justify each answer.With a pure strategy, a player makes the same choice each time the game is played.
Mark each statement True or False (T/F). Justify each answer.Each pure strategy is an optimal strategy.
Mark each statement True or False (T/F). Justify each answer.The value (x) of a particular strategy x to player R is equal to the maximum of the inner product of x with each of the columns of the payoff matrix.
Consider the matrix game where A has no saddle point.a. Find a formula for the optimal strategies x̂ for R and ŷ for C. What is the value of the game?b. Let and let α and β be real numbers with α ≠ 0. Use your answer in part (a) to show that the optimal strategies for the matrix game B =
Mark each statement True or False (T/F). Justify each answer.The value νR of the game to player R is the maximum of the values of the various possible strategies for R.
Mark each statement True or False (T/F). Justify each answer.The Fundamental Theorem for Matrix Games shows how to solve every matrix game.
Mark each statement True or False (T/F). Justify each answer.If column t dominates some other column in a payoff matrix A, then column t will not be used (that is, have probability zero) in an optimal strategy for (column) player C.
Let A be a matrix game having value v. Find an example to show that E(x, y) = ν does not necessarily imply that x and y are optimal strategies.
Let H be the hyperplane through the listed points. (a) Find a vector n that is normal to the hyperplane. (b) Find a linear functional f and a real number d such that H = [f : d]. 2 0 22 7
Suppose that the solutions of an equation Ax = b are all of the form x = x3u + p, where Find points v1 and v2 such that the solution set of Ax = b is aff {v1; v2}. u= 5 1 and p = -3 L 4 -2
Prove that the convex hull of a bounded set is bounded.
Prove that the open ball B(p,δ) = {x : ΙΙx – pΙΙ < δ}is a convex set.
Give an example of a compact set A and a closed set B in R2 such that (conv A) (convB) = ∅ but A and B cannot be strictly separated by a hyperplane.
Find the minimal representation of the polytope defined by the inequalities Ax ≤ b and x ≥ 0. A [1 10 = [₁ 2]. b = [1] 3 1 15
Determine if the set of points is affinely dependent. If so, construct an affine dependence relation for the points. 3 -2 252 3 5
Determine whether or not each set is compact and whether or not it is convex.Use the sets from Exercise 3.Data From Exercise 3Determine whether each set is open or closed or neither open nor closed. a. {(x, y):y> 0} b. {(x, y): x = 2 and 1 ≤ y ≤ 3} c. {(x, y): x = 2 and 1 < y < 3} d. {(x, y):
Find the minimal representation of the polytope defined by the inequalities Ax ≤ b and x ≥ 0. A = 2 3 1 4 b= = [ 18 16
Determine whether or not each set is compact and whether or not it is convex.Use the sets from Exercise 4.Data From Exercise 4Determine whether each set is open or closed or neither open nor closed. a. {(x, y): x² + y² = 1} b. {(x, y): x² + y² > 1} c. {(x, y): x² + y² ≤ 1 and y>0} d. {(x,
Determine if the set of points is affinely dependent. If so, construct an affine dependence relation for the points. [ ][][][] 0
Write y as an affine combination of the other points listed, if possible. 2 ----------] 2 V2 = -6 V3 = 3 7 VI
Determine whether each set is open or closed or neither open nor closed. a. {(x, y): x² + y² = 1} b. {(x, y): x² + y² > 1} c. {(x, y): x² + y² ≤ 1 and y>0} d. {(x, y): y ≥ x²} {(x, y): y < x²} e.
Repeat Exercise 2 where m is the minimum value of f on S instead of the maximum value.Data Form Exercise 2Given points in R2, let let S D conv {p1; p2; p3}. For each linear functional f, find the maximum value m of f on the set S, and find all points x in S at which f(x) = m.a. f (x1, x2) = x1 +
Write y as an affine combination of the other points listed, if possible. ----4----- [ V2 = -2 V3 = 6 --["}]] y 5
Determine whether each set is open or closed or neither open nor closed. a. {(x, y):y> 0} b. {(x, y): x = 2 and 1 ≤ y ≤ 3} c. {(x, y): x = 2 and 1 < y < 3} d. {(x, y): xy = 1 and x > 0} e. {(x, y): xy ≥ 1 and x >0}
Repeat Exercise 1 where m is the minimum value of f on S instead of the maximum value.Data From Exercise 1Given points in R2, let S = conv {p1; p2; p3}. For each linear functional f, find the maximum value m of f on the set S, and find all points x in S at which f (x) = m.a. f (x1,x2) = x1 –
Given points in R2,let let S D conv {p1; p2; p3}. For each linear functional f, find the maximum value m of f on the set S, and find all points x in S at which f(x) = m.a. f (x1, x2) = x1 + x2 b. f (x1, x2) = x1 – x2c. f (x1; x2) = 2x1 + x2 P₁ 0 = [-] P₂ = [2]. 1 1 , and p3 = 1 [2]
Given points in R2, let S = conv {p1; p2; p3}. For each linear functional f, find the maximum value m of f on the set S, and find all points x in S at which f (x) = m.a. f (x1,x2) = x1 – x2 b. f (x1, x2) = x1 + x2c. f (x1; x2) = 3x1 + x2 P = [6].P = [3], and ps = [-2] P2 P3
Determine if the set of points is affinely dependent. If so, construct an affine dependence relation for the points. 3 [³] [8] [3] 2 -3
Let B be an n x n symmetric matrix such that B2 = B. Any such matrix is called a projection matrix (or an orthogonal projection matrix). Given any y in Rn, let ŷ = By and z = y = ŷ.a. Show that z is orthogonal to ŷ.b. Let W be the column space of B. Show that y is the sum of a vector in W and
Let u be a unit vector in Rn, and let B = uuT.a. Given any x in Rn, compute Bx and show that Bx is the orthogonal projection of x onto u, as described in Section 6.2.b. Show that B is a symmetric matrix and B2 = B.c. Show that u is an eigenvector of B. What is the corresponding eigenvalue? Given a
Construct a spectral decomposition of A from Example 2. EXAMPLE 2 If possible, diagonalize the matrix A = 6-2-1 6-1 -2 -1 -1 5 TOT
Construct a spectral decomposition of A from Example 3. EXAMPLE 3 Orthogonally diagonalize the matrix A = characteristic equation is 3 -2 -2 6 4 2 0=- 0-1³ + 121²-212-98 (2-7)²(x + 2) = 43). whose
Let A and B be symmetric n x n matrices whose eigenvalues are all positive. Show that the eigenvalues of A + B are all positive.
Let A = PDP-1, where P is orthogonal and D is diagonal, and let λ be an eigenvalue of A of multiplicity k. Then λ appears k times on the diagonal of D. Explain why the dimension of the eigenspace for λ is k.
Let A be an n x n invertible symmetric matrix. Show that if the quadratic form xTAx is positive definite, then so is the quadratic form XTA-1x.
Suppose A = PRP–1, where P is orthogonal and R is upper triangular. Show that if A is symmetric, then R is symmetric and hence is actually a diagonal matrix.
Show that if an n x n matrix A is positive definite, then there exists a positive definite matrix B such that A = BTB. [Write A = PDPT, with PT = P-1. Produce a diagonal matrix C such that D = CTC, and let B = PCPT. Show that B works.
Suppose A is invertible and orthogonally diagonalizable. Explain why A–1 is also orthogonally diagonalizable.
Show that if B is m x n, then BTB is positive semidefinite; and if B is n x n and invertible, then BTB is positive definite.
Show how to classify a quadratic form Q(x) = xTAx, when and det A ≠ 0, without finding the eigenvalues of A.Verify the following statements: a. Qis positive definite if det A > 0 and a > 0. b. Q is negative definite if det A > 0 and a c. Q is indefinite if det A A = || a b b d
Suppose A is a symmetric n x n matrix and B is any n x m matrix. Show that BTAB, BTB, and BBT are symmetric matrices.
In mark each statement. Justify each answer.(T/F) The dimension of an eigenspace of a symmetric matrix is sometimes less than the multiplicity of the corresponding eigenvalue.
Concern an m x n matrix A with a reduced singular value decomposition, A = Ur DVrT, and the pseudoinverse A+ = Vr D–1UTr.Verify the properties of A+:a. For each y in Rm, AA+y is the orthogonal projection of y onto Col A.b. For each x in Rn, A+Ax is the orthogonal projection of x onto Row
Report the final matrix entries accurate to two decimal places. Use the method of Examples 3 and 4. A 1 6-8-4 7 0 -1 -8 -2 4 2 5 -4 -5 -6 4 2 2 4-8
Report the final matrix entries accurate to two decimal places. Use the method of Examples 3 and 4. A = -18 2 -14 -2 13 -4 4 19 -4 12 11 -12 21 ∞0 00 8 4 8
Construct the pseudoinverse of A. Begin by using a matrix program to produce the SVD of A, or, if that is not available, begin with an orthogonal diagonalization of ATA. Use the, pseudoinverse to solve Ax = b, for b = (6,-1,-4,6), and let x̂ be the solution. Make a calculation to verify that x̂
Show how to classify a quadratic form Q(x) = xTAx, when and det A ≠ 0, without finding the eigenvalues of A.If λ1 and λ2 are the eigenvalues of A, then the characteristic of A can be written in two ways: det(A - λI) and (λ – λ) (λ – λ2) Use this fact to show that λ1 + λ2 = a + d
Compute the singular values of the 4 x 4 matrix in Exercise 9 in Section 2.3, and compute the condition number σ1 = σ4.Data From Exercise 9 in Section 2.3Unless otherwise specified, assume that all matrices in these exercises are n x n. Determine which of the matrices in are invertible. Use as
In mark each statement. Justify each answer.(T/F) An n x n symmetric matrix has n distinct real eigenvalues.
In matrices are n x n and vectors are in Rn. Justify each answer.(T/F) If A is symmetric and the quadratic form xTAx has only negative values for x ≠ 0, then the eigenvalues of A are all positive.
In mark each statement. Justify each answer.(T/F) If AT = A and if vectors u and v satisfy Au = 3u and Av = 4v, then u · v = 0.
In matrices are n x n and vectors are in Rn. Justify each answer.(T/F) A Cholesky factorization of a symmetric matrix A has the form A = RTR, for an upper triangular matrix R with positive diagonal entries.
In mark each statement. Justify each answer.(T/F) If B = PDPT , where PT = P–1 and D is a diagonal matrix, then B is a symmetric matrix.
In matrices are n x n and vectors are in Rn. Justify each answer.(T/F) An indefinite quadratic form is neither positive semidefinite nor negative semidefinite.
Let and Verify that v1 and v2 are eigenvectors of A. Then orthogonally diagonalize A. A = 3 -1 1 3 -1 3 -1 1 -1 V₁ = 0 1
In matrices are n x n and vectors are in Rn. Justify each answer.(T/F) If the eigenvalues of a symmetric matrix A are all positive, then the quadratic form xTAx is positive definite.
A is an m x n matrix with a singular value decomposition A = UΣVT, where U is an m x m orthogonal matrix, Σ is an m x n "diagonal" matrix with r positive entries and no negative entries, and V is an n x n orthogonal matrix. Justify each answer.Using the notation of Exercise 23, show that ATuj =
In mark each statement. Justify each answer.(T/F) An n x n matrix that is orthogonally diagonalizable must be symmetric.
In matrices are n x n and vectors are in Rn. Justify each answer.(T/F) The principal axes of a quadratic form xTAx are eigenvectors of A.
In matrices are n x n and vectors are in Rn. Justify each answer.(T/F) A positive definite quadratic form Q satisfies Q(x) > 0 for all x in Rn.
a. Show that if A is positive definite, then A has an LU factorization, A = LU, where U has positive pivots on its diagonal.b. Show that if A has an LU factorization, A = LU, where U has positive pivots on its diagonal, then A is positive definite.c. Find an LU factorisation of and use it to obtain
If A is m x n, then the matrix G = ATA is called the Gram matrix of A. In this case, the entries of G are the inner products of the columns of A. a. Show that the Gram matrix of any matrix A is positive semidefinite, with the same rank as A.b. Show that if the columns of A are linearly
Let T : Rn → Rm be a linear transformation. Describe how to find a basis B for Rn and a basis C for Rm such that the matrix for T relative to B and C is an m x n “diagonal” matrix.
Prove that an n x n matrix A is positive definite if and only if A admits a Cholesky factorization, namely A = RTR for some invertible upper triangular matrix R whose diagonal entries are all positive.
In matrices are n x n and vectors are in Rn. Justify each answer.(T/F) A quadratic form has no cross-product terms if and only if the matrix of the quadratic form is a diagonal matrix.
Let Verify that 3 is an eigenvalue of A and v is an eigenvector. Then orthogonally diagonalize A . || -1 -1 5-1 5 -1 -1 -1 5 and v= [
A is an m x n matrix with a singular value decomposition A = UΣVT, where U is an m x m orthogonal matrix, Σ is an m x n "diagonal" matrix with r positive entries and no negative entries, and V is an n x n orthogonal matrix. Justify each answer.Let U = [u1.......um] and V = [v1 ....... un] ;
Orthogonally diagonalize the matrices in giving an orthogonal matrix P and a diagonal matrix D. To save you time, the eigenvalues in are the following: (17) –5, 5, 8; (18) 1, 2, 5; (19) 8, –1; (20) –3, 15; (21) 3, 5, 9; (22) 4, 6. 5 0 1 0 0 5 0 1 1 0 5 0 0 1 0 5

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