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linear algebra and its applications

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

Letand W = Span {u1, u2}.a. Let U = [u1 u2]. Compute UTU and UUT. b. Compute projw y and (UUT)y. 4 -[1] y = 8 U₁ = 2/3 1/3 2/3 U₂ = -2/3 2/3 1/3
All vectors and subspaces are inRn. Justify each answer.(T/F) The Gram-Schmidt process produces from a linearly independent set {x1,..., xp} an orthogonal set {v1,..., vp} with the property that for each k, the vectors v1,...,vk span the same subspace as that spanned by x1,..., xk.
All vectors are in Rn. Justify each answer.(T/F) v • v = ||v||2.
Determine which pairs of vectors are orthogonal. W= 3 -6 7 8 Z= -9 6 17 -7
Let {v1,.........,vp} be an orthonormal set. Verify the following equality by induction, beginning with p = 2. If x = c1v1 + .........+ cpvp, then ΙΙxΙΙ2 = Ιc1Ι2 +........+ ΙcpΙ2
True or False (T/F). Justify each answer. u, v, and w are vectors. If (u, v) = 0, then either u = 0 or v = 0.
Determine which pairs of vectors are orthogonal. 6 || -5 325 || B-A
Refer to vectors in Rn (or Rm) with the standard inner product. Justify each answer.(T/F) The normal equations for a least-squares solution of Ax = b are given by x̂ = (ATA)–1 ATb.
A is an m × n matrix and b is in Rm. Justify each answer.(T/F) A least-squares solution of Ax = b is a vector x̂ that satisfies Ax̂ = b̂, where b̂ is the orthogonal projection of b onto Col A.
All vectors and subspaces are inRn. Justify each answer.(T/F) If W = Span {X1, X2, X3} with {X1, X2, X3} linearly in- dependent, and if {V1, V2, V3} is an orthogonal set in W, then {V1, V2, V3) is a basis for W.
Let y, v1, and v2 be as in Exercise 12. Find the distance from y to the subspace of R4 spanned by v1 and v2.Data from in Exercises 12Find the closest point to y in the subspace W spanned by v1 and v2. y = 4 3 4 7 V1 = 1 V2 || 1 1
Refer to vectors in Rn (or Rm) with the standard inner product. Justify each answer.(T/F) A least-squares solution of Ax = b is the vector Ax̂ in Col A closest to b, so that ΙΙb - Ax̂ΙΙ ≤ ΙΙb - AxΙΙ for all x.
Use the factorization A = QR to find the least-squares solution of Ax = b. 4 = 3 نا بنا بنا لیا 3 3 3 500S = 1/2 1/27 - 1/2 1/2 - 1/2 1/2 1/2 1/2_ 6 0 55 . b = 9 -8 5 -3
All vectors and subspaces are inRn. Justify each answer.(T/F) If {v1, v2, v3} is an orthogonal basis for W, then multiplying v3 by a scalar c gives a new orthogonal basis {v1, v2, cv3}.
LetCompute the distance from y to the line through u and the origin. y || -1 7 and u = 3
A is an m × n matrix and b is in Rm. Justify each answer.(T/F) If b is in the column space of A, then every solution of Ax = b is a least-squares solution.
Determine which pairs of vectors are orthogonal. X = 4 -2 5 , y = 11 -1 -9
LetFind the distance from y to the plane in R3 spanned by u1 and u2. y ------ = -5 1 5 = -9 5 = -3 2 1
Refer to vectors in Rn (or Rm) with the standard inner product. Justify each answer.(T/F) If W is a subspace, then ΙΙprojw vΙΙ2 + ΙΙv - projw vΙΙ2 ΙΙvΙΙ2.
Use the inner product axioms and other results of this section to verify the statements in.||u + v||2 + ||u – v||2 = 2 ||u||2 + 2||v||2.
Use the inner product axioms and other results of this section to verify the statements.If {u, v} is an orthonormal set in V, then ||u-v|| = √2.
A is an m × n matrix and b is in Rm. Justify each answer.(T/F) The general least-squares problem is to find an x that makes Ax as close as possible to b.
Use the inner product axioms and other results of this section to verify the statements.(u, v) = 1/4 ||u + v||2 – 1/4||u – v||2.
LetCompute the distance from y to the line through u and the origin. y = 1 and u = 8 6
Use the factorization A = QR to find the least-squares solution of Ax = b. A = 2 2 1 3 341 = 2/3 2/3 -1/3 2/3 1/3-2/3 3 0 7 {].b = 2³ 3
Let f4 and f5 be the fourth-order and fifth-order Fourier approximations in C[0, 2π] to the square wave function in Exercise 10. Produce separate graphs of f4 and fs on the interval [0, 2π], and produce a graph of f5 on [-2π, 2π].Data Form Exercise 10Find the third-order Fourier
Determine which pairs of vectors are orthogonal. 8 -2 a = =[-] b = [3³] , -5 -3
Find the best approximation to z by vectors of the form c1v1 + c2v2. Z= 2 4 V₁ = 2 0 -3 V₂: = 5 -2 4 2
Refer to vectors in Rn (or Rm) with the standard inner product. Justify each answer.(T/F) If a square matrix has orthonormal columns, then it also has orthonormal rows.
The columns of Q were obtained by applying the Gram–Schmidt process to the columns of A. Find an upper triangular matrix R such that A = QR. Check your work. A = -2 3 5 7 2-2 4 6 Q = -2/7 5/7 5/7 2/7 2/7 -4/7 4/7 2/7
A certain experiment produces the data (1,7.9), (2, 5.4) and (3–.9). Describe the model that produces a least-squares fit of these points by a function of the formy = A cos x + B sin x
Find the distance between, 0 -[9]. 3 = n and z -7 -5 7
A simple curve that often makes a good model for the variable costs of a company, as a function of the sales level x, has the form y = β1x +β2x2 + β3x3.There is no constant term because fixed costs are not included.a. Give the design matrix and the parameter vector for the linear model that
The columns of Q were obtained by applying the Gram–Schmidt process to the columns of A. Find an upper triangular matrix R such that A = QR. Check your work. A = 5 1 -3 1 9 7 -5 5 Q = 5/6 -1/6 1/6 5/6 1/6 3/6 -3/6 1/6
Suppose the first few Fourier coefficients of some function f in C[0,2π] are a0, a1, a2, and b1, b2, b3. Which of the following trigonometric polynomials is closer to f? Defend your answer. g(t)= +a, cost + a₂ cos 2t + b₁ sint ao 2 h(t)= ao + a cost + a₂ cos 2t + by sint + b₂ sin 21 2
LetCompute Au and Av, and compare them with b. Is it possible that at least one of u or v could be a least-squares solution of Ax = b? (Answer this without computing a least squares solution.) A = 2 -3 5 -|------ = 3 2 = 4 -5 - = [-]. and v=
Use the inner product axioms and other results of this section to verify the statements.(u, cv) = c(u, v) for all scalars c.
Find the best approximation to z by vectors of the form c1v1 + c2v2. Z= 3 -7 2 3 - V₁ = 2 1 , V₂ = 1 0
Compute the orthogonal projection ofonto the line throughand the origin. -3 [+] 4
Refer to the data in Exercise 19 in Section 6.6, concerning the takeoff performance of an airplane. Suppose the possible measurement errors become greater as the speed of the airplane increases, and let W be the diagonal weighting matrix whose diagonal entries are 1, 1, 1, .9, .9, .8, .7, .6, .5,
LetWrite y as the sum of a vector in Span {u} and a vector orthogonal to u. y || 2 [²] 6 L and u = [9]
Refer to vectors in Rn (or Rm) with the standard inner product. Justify each answer.(T/F) A square matrix with orthogonal columns is an orthogonal matrix.
A certain experiment produces the data (1,2.5), (2,4.3), (3,5.5), (4,6.1), (5,6.1). Describe the model that produces a least-squares fit of these points by a function of the form y = β1x + β2x2 Such a function might arise, for example, as the revenue from the sale of x units of a product,
Find (a) The orthogonal projection of b onto Col A .(b) A least-squares solution of Ax = b. A = 1 2 -1 0 12 -1 1 0 2 - 1 0 b = 3 9 9 3
LetWrite y as the sum of two orthogonal vectors, one in Span {u} and one orthogonal to u. У 2 3 and u = |- 4 -7
Find an orthogonal basis for the column space of each matrix. 12 4 -3 -3 1 1 - -1 0 -1 1 -1 2 4
LetCompute Au and Av, and compare them with b. Could u possibly be a least-squares solution of Ax = b? 3 4 A = -2 1 b -- |- |- - - - - - - - - - -}- _$] -9 5 and V 3 4
Let T be a one-to-one linear transformation from a vector space V into Rn. Show that for u, v in V, the formula (u, v) = T(u) · T(v) defines an inner product on V.
Find a unit vector in the direction of the given vector. 3/3 [8/³] 1
Find the closest point to y in the subspace W spanned by v1 and v2. y = 4 3 4 7 V1 = 1 V2 || 1 1
Refer to vectors in Rn (or Rm) with the standard inner product. Justify each answer.(T/F) If a matrix U has orthonormal columns, then UUT = 1.
Let X be the design matrix in Example 2 corresponding to a least-squares fit of a parabola to data (x1; y1),.......,(xn,yn). Suppose x1, x2, and x3 are distinct. Explain why there is only one parabola that fits the data best, in a least-squares sense. EXAMPLE 2 If a machine learns the data from
Compute the quantities using the vectors. 3 6 U= --[+] ·-() --[]· --[-] = [3]· W= X = -2 -5 3
Explain why a Fourier coefficient of the sum of two functions is the sum of the corresponding Fourier coefficients of the two functions.
Refer to vectors in Rn (or Rm) with the standard inner product. Justify each answer.(T/F) If {v1, v2, v3} is an orthogonal set and if c1, c2, and c3 are scalars, then {c1v1, c2 v2, c3 v3) is an orthogonal set.
Compute the least-squares error associated with the least squares solution found in Exercise 4.Data from in Exercise 4Find a least-squares solution of Ax = b by 1 1 ^-+-+-+- A = 1 -4, b= 1 1 9 2 5
Let A be any invertible n x n matrix. Show that for u, v in Rn, the formula (u, v) = (Au). (Av) = (Au)T(Av) defines an inner product on Rn.
The space is C [0; 2π] with the inner product (6).Find the third-order Fourier approximation to sin3t, without performing any integration calculations.
Compute the least-squares error associated with the least squares solution found in Exercise 3.Data from in Exercise 3Find a least-squares solution of Ax = b by A = 1-2 2 3 5 -1 0 2 b || 1 -4 2
Compute the quantities using the vectors. 3 6 U= --[+] ·-() --[]· --[-] = [3]· W= X = -2 -5 3
Find a polynomial p3 such that {p0, p1, p2, p3) is an orthogonal basis for the subspace p3 of p4. Scale the polynomial p3 so that its vector of values is (-1,2,0, 2, 1).
You may assume that {u1,..., u4} is an orthogonal basis for R4.Write v as the sum of two vectors, one in Span {u1} and the other in Span {u2, u3, u4}. = 'n , U₂ = , U3 = = 'n 3
Find a least-squares solution of Ax = b by (a) Constructing the normal equations for x̂ (b) Solving for x̂. A: 2 -2 2 1 0 |]+= [ b 3 -5 8 1
Refer to vectors in Rn (or Rm) with the standard inner product. Justify each answer.(T/F) If W is a subspace of Rn, then W and W⊥ have no vectors in common.
Refer to vectors in Rn (or Rm) with the standard inner product. Justify each answer.(T/F) If ΙΙu – vΙΙ2= ΙΙuΙΙ2 + ΙΙvΙΙ2, then u and v are orthogonal.
Refer to vectors in Rn (or Rm) with the standard inner product. Justify each answer.(T/F) If ΙΙu + vΙΙ2 = ΙΙuΙΙ2 + ΙΙvΙΙ2, then u and v are orthogonal.
Refer to vectors in Rn (or Rm) with the standard inner product. Justify each answer.(T/F) The orthogonal projection of y onto u is a scalar multiple of y.
The space is C [0; 2π] with the inner product (6).Show that sinmt and cos nt are orthogonal for all positive integers m and n.
Determine which sets of vectors are orthogonal. OD -1 4 -3 5 2 1 3 -4 -7
Compute the quantities using the vectors. 3 6 U= --[+] ·-() --[]· --[-] = [3]· W= X = -2 -5 3
The given set is a basis for a subspace W. Use the Gram–Schmidt process to produce an orthogonal basis for W. 0 ∞ 5 -6
Let R2 have the inner product of Example 1. Show that the Cauchy–Schwarz inequality holds for x = .(3,– 4) and y = (–4; 3). EXAMPLE 1 Fix any two positive numbers-say, 4 and 5-and for vectors u = (₁, ₂) and v = (v₁, v₂) in R², set (u, v) = 4u1v₁ + 5u2v2 Show that equation (1)
You may assume that {u1,..., u4} is an orthogonal basis for R4.Write x as the sum of two vectors, one in Span {u1, u2, u3} and the other in Span {u4}. --0-0-0-0-0 = = 1 = 3 5 = 5 -3 X = 10 -8 2
P2 with the inner product given by evaluation at –1, 0, and 1.Compute ||p|| and || q ||, for p and q in Exercise 4.Data From Exercise 4P2 with the inner product given by evaluation at –1, 0, and 1.Compute (p, q), where p(t) = 4t - 3t2, q(t) = 1 + 9t2.
P2 with the inner product given by evaluation at –1, 0, and 1.Compute the orthogonal projection of q onto the subspace spanned by p, for p and q in Exercise 3.Data From ExerciseP2 with the inner product given by evaluation at –1, 0, and 1.Compute (p, q), where p(t) = 4 + t, q(t) = 5 –4t2.
The space is C [0; 2π] with the inner product (6).Show that sin mt and sin nt are orthogonal when m ≠ n.
Find the equation y = β0 + β1 x of the least squares line that best fits the given data points.(1,0),(2,2),(3,7),(4,9)
Refer to vectors in Rn (or Rm) with the standard inner product. Justify each answer.(T/F) If x is orthogonal to both u and v, then x must be orthogonal to u - V.
Refer to vectors in Rn (or Rm) with the standard inner product. Justify each answer.(T/F) A vector v and its negative, –v, have equal lengths.
If a machine learns the least-squares line that best fits the data in Exercise 1, what will the machine pick for the value of y when x = 4?Data From Exercise 1Find the equation y = β0 + β1 x of the least squares line that best fits the given data points.(0, 1),(1,1),(2, 2),(3, 2)
Find the least-squares line y = β0 + β1 x that best fits the data (-2,0), (-1,0), (0, 2), (1,4), and (2,4), assuming that the first and last data points are less reliable. Weight them half as much as the three interior points.
Suppose 5 out of 25 data points in a weighted least-squares problem have a y-measurement that is less reliable than the others, and they are to be weighted half as much as the other 20 points. One method is to weight the 20 points by a factor of 1 and the other 5 by a factor of 1/2. A second method
Find the equation y = β0 + β1 x of the least squares line that best fits the given data points.(0, 1), (1,1), (2, 2), (3, 2)
Refer to vectors in Rn (or Rm) with the standard inner product. Justify each answer.(T/F) The length of every vector is a positive number.
Find a least-squares solution of Ax = b by (a) Constructing the normal equations for x̂ (b) Solving for x̂. A = -1 2 2-3, b 4 --0 1 2 -1 3
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.Show that if A is square, then |det Al is the product
Find(a) The maximum value of Q(x) subject to the constraint xTx = 1, (b) A unit vector u where this maximum is attained, and (c) The maximum of Q(x) subject to the constraints xTx = 1 and xTu = 0.-6x12 - 10x22 - 13x23 - 13x24 - 4x1x2 - 4x1 x3 - 4x1x4 +
Mark each statement True or False. Justify each answer. In each part, A represents an n x n matrix.If A is n x n, then A and ATA have the same singular values.
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: 5 -4 -4 11
Repeat Exercise 15 for the following SVD of a 3 x 4 matrix A:Data From Exercise 15Suppose the factorization below is an SVD of a matrix A, with the entries in U and V rounded to two decimal places.a. What is the rank of A?b. Use this decomposition of A, with no calculations, to write a basis for
Find(a) The maximum value of Q(x) subject to the constraint xTx = 1, (b) A unit vector u where this maximum is attained, and (c) The maximum of Q(x) subject to the constraints xTx = 1 and xTu = 0.4x12 - 6x1x2 - 10x1x3 - 10x1x4 - 6x2x3 - 6x2x4 - 2x3x4
Mark each statement True or False. Justify each answer. In each part, A represents an n x n matrix.A singular value decomposition of an m x n matrix B can be written as B = PΣQ, where P is an m x m orthogonal matrix, Q is an n x n orthogonal matrix, and is an m x n "diagonal" matrix.
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: 5 6 6 10
Suppose the factorization below is an SVD of a matrix A, with the entries in U and V rounded to two decimal places.a. What is the rank of A?b. Use this decomposition of A, with no calculations, to write a basis for ColA and a basis for NulA. A = X ][ .40 -.78 .37.33.87 -.84.52.16 .47 .30.51.81 .76
Justify each answer. In each part, A represents an n x n matrix.True or False. If B is m x n and x is a unit vector in Rn, then is the first singular value of B. ΙΙBxΙΙ ≤ σ1, where is the first singular value of B.
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: 2-3 2 -3 L
Find(a) The maximum value of Q(x) subject to the constraint xTx = 1, (b) A unit vector u where this maximum is attained, and (c) The maximum of Q(x) subject to the constraints xTx = 1 and xTu = 0.3x1x2 + 5x1x3 + 7x1x4 + 7x2x3 + 5x2x4 + 3x3x4
Then make a change of variable, x = Py that transform the quadratic form into one with no cross-product terms. Write the new quadratic form. Construct P using the methods of Section 7.1. 5x² + 12x1x2
In Exercise 7, find a unit vector x at which Ax has maximum length.Data From Exercise 7Find an SVD of each matrix. 22 -1 2
Justify each answer. In each part, A represents an n x n matrix.True or False. If U is m x n with orthogonal columns, then UUTx is the orthogonal projection of x onto Col U.

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