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mathematics
linear algebra and its applications

Questions and Answers of Linear Algebra And Its Applications

Justify each answer. In each part, A represents an n x n matrix.True or False. If P is an n x n orthogonal matrix, then the change of variable x = Pu transforms xTAx into a quadratic form whose
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: 4 1 4
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. x² -
Find the SVD of . A = 3 2 2 2 3-2
Let A be an n x n symmetric matrix, let M and m denote the maximum and minimum values of the quadratic form xTx, where xTx = 1, and denote corresponding unit eigenvectors by u1 and un. The
The sample covariance matrix is a generalization of a formula for the variance of a sample of N scalar measurements, say, t1...........tN . If m is the average of t1...........tN , then the sample
Justify each answer. In each part, A represents an n x n matrix.True or False. An indefinite quadratic form is one whose eigenvalues are not definite.
Given multivariate data X1,..., XN (in RP) in mean- deviation form, let P be a p x p matrix, and define Yk = PT Xk for k = 1,..., N. a. Show that Y1,..., YN are in mean-deviation
Suppose x is a unit eigenvector of a matrix A corresponding to an eigenvalue 3. What is the value of xTAx?
Find an SVD of each matrix.One choice for U is 1 -2 6-2 -3 6
The given set is a basis for a subspace W. Use the Gram–Schmidt process to produce an orthogonal basis for W. -2 1 -2 56 5 -6 7 -8
Determine if the matrix pairs are controllable. A = 0 1 0 0 0 0 -1 -13 0 1 0 -12.2 0 |--B , B = 0 0 0 1 -1.5
A laboratory animal may eat any one of three foods each day. Laboratory records show that if the animal chooses one food on one trial, it will choose the same food on the next trial with a
A small remote village receives radio broadcasts from two radio stations, a news station and a music station. Of the listeners who are tuned to the news station, 70% will remain listening to the news
Let B = {b1, b2, b3} be a basis for the vector space V . Let T: V →V be a linear transformation with the property thatFind [T]B, the matrix for T relative to B. T(b₁) = 3b₁5b2, T (b₂) = -b₁
Is λ = - 2 an eigenvalue ofWhy or why not? 7 3 3-1 ?
Let B = {b1, b2} be a basis for a vector space V . Let T: V → V be a linear transformation with the property thatFind [T]B, the matrix for T relative to B. T(b₁) = 7b₁ + 4b2, T (b₂) = 6b₁ -
The matrix A is followed by a sequence {xk} produced by the power method. Use these data to estimate the largest eigenvalue of A, and give a corresponding eigenvector. 5 ^= [1
Find the characteristic polynomial and the eigenvalues of the matrices. 88 4 4 8
On any given day, a student is either healthy or ill. Of the students who are healthy today, 95% will be healthy tomorrow. Of the students who are ill today, 55% will still be ill tomorrow.a. What is
Let each matrix in act on C2. Find the eigenvalues and a basis for each eigenspace in C2. -1 5 - 1 S-
Find the characteristic polynomial and the eigenvalues of the matrices. 3-2 -1
Assume the mapping T: P2 → P2 defined byis linear. Find the matrix representation of T relative to the basis B = {1, t, t}. T (ao + a₁ + a₂t²) = 2 ao +(3 a₁ + 4a₂)t + (5 ao − 6a₂)t²
Isan eigenvector ofIf so, find the eigenvalue. 4
Use the factorization A = PDP-1 to compute Ak, where k represents an arbitrary positive integer. 1 a 0 [3a-6 ]-[ ][ ][-1 = [3 0 b b) 0 b -3
Justify each answer.(T/F) If A is row equivalent to the identity matrix I, then A is diagonalizable.
Let each matrix in act on C2. Find the eigenvalues and a basis for each eigenspace in C2. 1 2 -4 5
Justify each answer.(T/F) If A contains a row or column of zeros, then 0 is an eigenvalue of A.
The matrix A is followed by a sequence {xk} produced by the power method. Use these data to estimate the largest eigenvalue of A, and give a corresponding eigenvector. A -[: .5 .2 .4 .7 [1] [3]
The weather in Edinburgh is either good, indifferent, or bad on any given day. If the weather is good today, there is a 50% chance the weather will be good tomorrow, a 30% chance the weather will be
Justify each answer.(T/F) Each eigenvalue of A is also an eigenvalue of A2.
Find the characteristic polynomial and the eigenvalues of the matrices. 5 -5 -2 نا
Define T: P2 → P2 by T(p) = p(0) - p(1)t + p(2)t2. a. Show that I is a linear transformation. b. Find T (p) when p(t) = -2 + t. Is p an eigenvector of T? c. Find the matrix for T
Use the factorization A = PDP-1 to compute Ak, where k represents an arbitrary positive integer. 15 6 -36 -15 23 -3 0 -1 = [2 ³ ][ ³][3][4] I 1 1 03 1 -2
Let each matrix in act on C2. Find the eigenvalues and a basis for each eigenspace in C2. -7 -5 -3
The matrix A is followed by a sequence {xk} produced by the power method. Use these data to estimate the largest eigenvalue of A, and give a corresponding eigenvector. 4.1 6 ^=[43
LetFind a vector with a 1 in the second entry that is close to an eigenvector of A. Use four decimal places. Check your estimate, and give an estimate for the dominant eigenvalue of A. 15 16 A =[₁
Find the steady-state vector. .7.1 .2
Find the B-matrix for the transformation x ↦ Ax, when B = {b1, b2}. 4 9 4= [1 2])-[1]=[] A ,b₂ 4
Justify each answer.(T/F) Eigenvalues must be nonzero scalars.
If a machine learns the least-squares line that best fits the data in Exercise 2, what will the machine pick for the value of y when x = 3?Data From Exercise 2Find the equation y = β0 + β1 x
Determine which sets of vectors are orthogonal. 5 -4 0 3 لیا -4 1 -3 8 33 3 5 -1
Verify that {u1, u2} is an orthogonal set, and then find the orthogonal projection of y onto Span {u1, u2}. y = ----] = -1 5, U₁ 3 = -5
Isan eigenvector ofIf so, find the eigenvalue.
Find the characteristic polynomial and the eigenvalues of the matrices. 5 -4 3 4
Determine which sets of vectors are orthogonal. 3 -2 1 3 -1 3 -3 4 3 8 7 0
The given set is a basis for a subspace W. Use the Gram–Schmidt process to produce an orthogonal basis for W. 1 -4 0 7 -7 -4 1
The given set is a basis for a subspace W. Use the Gram–Schmidt process to produce an orthogonal basis for W. 5 6
Verify that {u1, u2} is an orthogonal set, and then find the orthogonal projection of y onto Span {u1, u2}. y = 3 [-----] = , 2 2 6 =
Compute the quantities using the vectors. 3 6 U= --[+] ·-() --[]· --[-] = [3]· W= X = -2 -5 3
Determine which sets of vectors are orthogonal. -5 HOD -2 1 -2 1 2 1
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 5 -1 0 2 b || 1 -4 2
Compute the quantities using the vectors. 3 6 U= --[+] ·-() --[]· --[-] = [3]· W= X = -2 -5 3
Determine which sets of vectors are orthogonal. 2 ]] [ 0 0 -5 -3 4 2 6
The given set is a basis for a subspace W. Use the Gram–Schmidt process to produce an orthogonal basis for W. 4 -5 6 -8 17 -19
Verify that {u1, u2} is an orthogonal set, and then find the orthogonal projection of y onto Span {u1, u2}. y = 3 [---0--[] 3 U₁ 4 = 3 4 -2 -4
Find the equation y = β0 + β1 x of the least squares line that best fits the given data points.(2,3),(3,2),(5,1),(6,0)
Compute the quantities using the vectors. 3 6 U= --[+] ·-() --[]· --[-] = [3]· W= X = -2 -5 3
Find a least-squares solution of Ax = b by (a) Constructing the normal equations for x̂ (b) Solving for x̂. 1 1 ^-+-+-+- A = 1 -4, b= 1 1 9 2 5
The given set is a basis for a subspace W. Use the Gram–Schmidt process to produce an orthogonal basis for W. 2 -5 1 4 北 -1 2
Verify that {u1, u2} is an orthogonal set, and then find the orthogonal projection of y onto Span {u1, u2}. y = 4 3 , u₁ = 1 U₂ = 1 0
Find the equation y = β0 + β1 x of the least squares line that best fits the given data points.(–1,0),(0,1),(1,2),(2,4)
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 4.Data From Exercise 4P2 with the inner
Show that {u1, u2} or {u1, u2, u3} is an orthogonal basis for R2 or R3, respectively. Then express x as a linear combination of the u's. U₁ = [³] ₁ ₂ = [-²] and x = [ - ] -4 3 U12 6
let W be the subspace spanned by the u’s, and write y as the sum of a vector in W and a vector orthogonal to W. y = [D--0--B 6 4 , 4 -3
If a machine learns the least-squares line that best fits the data in Exercise 2, what will the machine pick for the value of y when x = 4? How closely does this match the data point at x = 4 fed
Show that {u1, u2} or {u1, u2, u3} is an orthogonal basis for R2 or R3, respectively. Then express x as a linear combination of the u's. 2 ₁ = [-²]₁ ₂ = [2], and x = [-2] 12 -3 -7
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 = 3? How closely does this match the data point at x = 3 fed
let W be the subspace spanned by the u’s, and write y as the sum of a vector in W and a vector orthogonal to W. ------] 3 -2 y = 3 5 5 = 1 4
Compute the quantities using the vectors. 3 6 U= --[+] ·-() --[]· --[-] = [3]· W= X = -2 -5 3
Show that ||cos kt||2 = π and ||sinkt||2 = π for k > 0.
Compute the quantities using the vectors. 3 6 U= --[+] ·-() --[]· --[-] = [3]· W= X = -2 -5 3
Find the third-order Fourier approximation to f(t) = t – 1.
Find (a) The orthogonal projection of b onto Col A .(b) A least-squares solution of Ax = b. 1 3 -2 A = -[ 5 1 , b 4 4 -2 -3
Let P3 have the inner product given by evaluation at -3,-1, 1, and 3. Let p0(t) = 1, p1(t) = t, and p2(t) = t2.a. Compute the orthogonal projection of P2 onto the sub - space spanned by
Find a unit vector in the direction of the given vector. -30 40
Find the third-order Fourier approximation to f(t) = 2π – t.
If you enter the data from Exercise 1 into a machine and it returns a y value of 20 when x = 2:5, should you trust the machine? Justify your answer.Data From Exercise 1Find the equation y =
Let W be the subspace spanned by the u’s, and write y as the sum of a vector in W and a vector orthogonal to W. y = 4 3 3 -1 , u₁ = , 1₂ = 3 1 -2 U13 = , 0 1 1
Find an orthogonal basis for the column space of each matrix. 3 1 - 1 3 -5 1 1 5-2 8 -7
Show that {u1, u2} or {u1, u2, u3} is an orthogonal basis for R2 or R3, respectively. Then express x as a linear combination of the u's. --0--0--0--0 = 2 and = 4 4 6 4
Find (a) The orthogonal projection of b onto Col A .(b) A least-squares solution of Ax = b. A= 1 1 21-4-| 4, b 3 -1 5
Let P3 have the inner product as in Exercise 9, with p0, p1, and q the polynomials described there. Find the best approximation to p(t) = t3 by polynomials in Span {p0, p1. q}.Data From
Find a unit vector in the direction of the given vector. 3 6 -3
Find the third-order Fourier approximation to the square wave function f(t) = 1 for 0 ≤ t < π and f(t) = –1 for π ≤ t < 2π.
If you enter the data from Exercise 2 into a machine and it returns a y value of –1 when x = 1:5, should you trust the machine? Justify your answer.Data From Exercise 2Find the equation y =
Let W be the subspace spanned by the u’s, and write y as the sum of a vector in W and a vector orthogonal to W. y = 3 4 5 4 , u₁ = 1 0 -1 U₂ = 0 1 , U3 = 1
Find an orthogonal basis for the column space of each matrix. -1 6 3-8 1 1 6 3 -2 6 -4 -3
Find an orthogonal basis for the column space of each matrix. 12 5 1-4 4 4-3 -4 12 -1 - 1 1 7 1
Find the closest point to y in the subspace W spanned by v1 and v2. 3 1 -----| H = y = 3 1 5 V2 = 1
Compute the orthogonal projection ofonto the line throughand the origin. 7
Show that {u1, u2} or {u1, u2, u3} is an orthogonal basis for R2 or R3, respectively. Then express x as a linear combination of the u's. 2 --------] = 2 = = 4 0 4 and x = 3 7
Find (a) The orthogonal projection of b onto Col A.(b) A least-squares solution of Ax = b. A = 1 3 5 -1 -4 -4 1 0 1 10 ,b. b = 3 بنا -2 -4 7
Let p0, p1, and p2 be the orthogonal polynomials described in Example 5, where the inner product on P4 is given by eval-uation at -2, -1, 0, 1, and 2. Find the orthogonal projection of t3 onto Span
Find a unit vector in the direction of the given vector. 2/9 1/3 1
Refer to vectors in Rn (or Rm) with the standard inner product. Justify each answer.(T/F) The set of all vectors in Rn orthogonal to one fixed vector is a subspace of Rn.
Let X be the design matrix used to find the least-squares line to fit data (x1y1),.....(xn,yn). Use a theorem in Section 6.5 to show that the normal equations have a unique solution if and only if
Find the third-order Fourier approximation to cos2t, without performing any integration calculations.
Find the steady-state vector. .1 .9 .6 .4
Let B = {b1, b2, b3} be a basis for a vector space V. Find T(2b1 - 5b3) when T is a linear transformation from V to V whose matrix relative to B is [T]B = 1 0-4 2 0 2 -3 3

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