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mathematics
linear algebra

Linear Algebra A Modern Introduction 3rd edition David Poole - Solutions

The given vectors form a basis for R2or R3. Apply the Gram-Schmidt Process to obtain an orthogonal basis. Then normalize this basis to obtain an orthonormal basis. 2 X1 э X2 -2
The given vectors form a basis for R2or R3. Apply the Gram-Schmidt Process to obtain an orthogonal basis. Then normalize this basis to obtain an orthonormal basis. Н0 X1 |, X2 2
Let W be a subspace of Rn, and let x be a vector in Rn.Prove that x is in W if and only if projW(x) = x.
Let W be a subspace of Rn and v a vector in Rn. Suppose that w and w′ are orthogonal vectors with w in W and that v = w + w′. Is it necessarily true that w′ is in W⊥? Either prove that it is true or find a counterexample.
Prove Theorem 5.9(d).d. If W = span(w1, . . . , wk), then v is in W⊥ if and only if v · wi = 0 for all i 1, . . . , k.
Prove Theorem 5.9(c).c. W ⋂ W⊥= {0}
Find the orthogonal decomposition of v with respect to W. 2 -1 W = span| .3
Find the orthogonal decomposition of v with respect to W. W = span -2 2 3
Find the orthogonal projection of v onto the subspace W spanned by the vectors ui. (You may assume that the vectors uiare orthogonal.) -1 -2 > и -3 , и, %3D > Из —
Find the orthogonal projection of v onto the subspace W spanned by the vectors ui. (You may assume that the vectors uiare orthogonal.) 1 -1 2 , u1 3
Let W be the subspace spanned by the given vectors. Find a basis for WŠ¥. 3 3 -2 W1 0 , W2 -2 0 , w, = , W3 -1 -2 5
Find bases for the column space of A and the null space of ATfor the given exercise. Verify that every vector in col(A) is orthogonal to every vector in null(AT).Exercise 8Data from Exercise 8 - 1 4 0 -2 0 2 2 -2 3 2 4 5 1
Find bases for the row space and null space of A. Verify that every vector in row(A) is orthogonal to every vector in null(A). -2 0 4 0 |A = -2 3 2 2 1 2 4 2 5
Find the orthogonal complement WŠ¥of W and give a basis for WŠ¥. х у :х 3D 2t, у — — t, z %3D —1 2
Find the orthogonal complement WŠ¥of W and give a basis for WŠ¥. = –t, z = 3t y:x = t, y
Find the orthogonal complement WŠ¥of W and give a basis for WŠ¥. х 5z = 0 у : -х + Зу — 5z 3D 0
Find the orthogonal complement WŠ¥of W and give a basis for WŠ¥. х y:x+ y – z = 0
Find the orthogonal complement WŠ¥of W and give a basis for WŠ¥. = 0 : 3x + 2y
Find the orthogonal complement WŠ¥of W and give a basis for WŠ¥. х : 2х — у %3D 0
Let B = {v1,......, vn} be an orthonormal basis for Rn.(a) Prove that, for any x and y in Rn, x • y = (x • v1) (y • v1) + (x • v2) (y • v2) + ...........+ (x • vn) (y • vn) (This identity is called Parseval’s Identity.)(b) What does Parseval’s Identity imply about the relationship
Let x be a unit vector in Rn. Partition x asLet
Use Exercise 28 to determine whether the given orthogonal matrix represents a rotation or a reflection. If it is a rotation, give the angle of rotation; if it is a reflection, give the line of reflection.Data From Exercise 28(a) Prove that an orthogonal 2 Ã— 2 matrix must have the
Use Exercise 28 to determine whether the given orthogonal matrix represents a rotation or a reflection. If it is a rotation, give the angle of rotation; if it is a reflection, give the line of reflection.Data From Exercise 28(a) Prove that an orthogonal 2 Ã— 2 matrix must have the
Use Exercise 28 to determine whether the given orthogonal matrix represents a rotation or a reflection. If it is a rotation, give the angle of rotation; if it is a reflection, give the line of reflection.Data From Exercise 28(a) Prove that an orthogonal 2 Ã— 2 matrix must have the
Use Exercise 28 to determine whether the given orthogonal matrix represents a rotation or a reflection. If it is a rotation, give the angle of rotation; if it is a reflection, give the line of reflection.Data From Exercise 28(a) Prove that an orthogonal 2 Ã— 2 matrix must have the
(a) Prove that an orthogonal 2 Ã— 2 matrix must have the formwhereis a unit vector.(b) Using part (a), show that every orthogonal 2 Ã— 2 matrix is of the formwhere 0 ‰¤ u ‰¤ 2Ï€.(c) Show that every orthogonal 2 Ã— 2 matrix
If Q is an orthogonal matrix, prove that any matrix obtained by rearranging the rows of Q is also orthogonal.
Prove Theorem 5.8(d).d. If Q1 and Q2 are orthogonal nn matrices, then so is Q1Q2.
Prove Theorem 5.8(b).b. det Q = ±1
Prove Theorem 5.8(a).
Determine whether the given matrix is orthoglonal. If it is, find its inverse. 1/V6 1/V6 0 -2/3 1/V2 -1/V6 1/V2 2/3 1/V2 1/3
Determine whether the given matrix is orthoglonal. If it is, find its inverse. HIN -IN -IN -I2 HIN -IN 12 1/2 HIN -IN -IN -IN HIN -IN -IN -I2
Determine whether the given matrix is orthoglonal. If it is, find its inverse. - sin? 0 cos 0 sin 0 –cos 0 - cos 0 sin 0 cos? 0 sin 0 sin 0 cos 0
Determine whether the given matrix is orthoglonal. If it is, find its inverse. 1/3 l3/ H/N -I2-12
Determine whether the given matrix is orthoglonal. If it is, find its inverse. _1
Determine whether the given orthogonal set of vectors is orthonormal. If it is not, normalize the vectors to form an orthonormal set. V3/2 - V3/6 1/2 1/2 -1/2 1/2. V6/3 1/V6 -1/V6 V3/6 1/V2 - V3/6. L1/V2
Determine whether the given orthogonal set of vectors is orthonormal. If it is not, normalize the vectors to form an orthonormal set. HIN -IO -16 -6 HIN -IN -IN -2
Determine whether the given orthogonal set of vectors is orthonormal. If it is not, normalize the vectors to form an orthonormal set. 2. 2.
Show that the given vectors form an orthogonal basis for R2or R3. Then use Theorem 5.2 to express w as a linear combination of these basis vectors. Give the coordinate vector [w]Bof w with respect to the basis B = {v1, v2} of R2or B = {v1, v2, v3} of R3. 1 V1 -1 V2 V3 = -1 3 -1
Show that the given vectors form an orthogonal basis for R2or R3. Then use Theorem 5.2 to express w as a linear combination of these basis vectors. Give the coordinate vector [w]Bof w with respect to the basis B = {v1, v2} of R2or B = {v1, v2, v3} of R3. 0 |, v, = V1 -1 ]; w = 1 -1
Show that the given vectors form an orthogonal basis for R2or R3. Then use Theorem 5.2 to express w as a linear combination of these basis vectors. Give the coordinate vector [w]Bof w with respect to the basis B = {v1, v2} of R2or B = {v1, v2, v3} of R3. -2 3 V2 V1 6.
Show that the given vectors form an orthogonal basis for R2or R3. Then use Theorem 5.2 to express w as a linear combination of these basis vectors. Give the coordinate vector [w]Bof w with respect to the basis B = {v1, v2} of R2or B = {v1, v2, v3} of R3. 4 V2 V1 -3 -2
Determine which sets of vectors are orthogonal. -1 -1 -1 -1
Determine which sets of vectors are orthogonal. -2 -4 3 -6 -1 -1 2 4 7.
Determine which sets of vectors are orthogonal. 4 -2 6. 3 -2 -1
Determine which sets of vectors are orthogonal. -2 5 -2 2.
Find the general solution to the given system of differential equations. Then find the specific solution that satisfies the initial conditions. (Consider all functions to be functions of t.)x′ = y - z, x(0) = 1y′ = x + z, y(0) = 0z′ = x + y, z(0) = -1
Write out the first six terms of the sequence defined by the recurrence relation with the given initial conditions.b0 = 1, b1 = 1, bn = 2bn-1 + bn-2 for n ≥ 2
Let A be a non-negative, irreducible matrix such that I - A is invertible and (I - A)-1 ≥ O. Let λ1 and v1 be the Perron root and Perron eigenvector of A.(a) Prove that .(b) Deduce from (a) that v1 Av1.
Prove that a 2 Ã— 2 matrixis reducible if and only if a12 = 0 or a21 = 0. A12 A = a22 a21 И22
Let A and B be n × n matrices, x a vector in Rn, and c a scalar. Prove the following matrix inequalities:a.|cA| = |c| |A|b.|A + B| ≤ |A| + |B|c.|Ax| ≤ |A| |x|d.|AB| ≤ |A| |B|
A graph is called k-regular if k edges meet at each vertex. Let G be a k-regular graph.(a) Show that the adjacency matrix A of G has λ = k as an eigenvalue.(b) Show that if A is primitive, then the other eigenvalues are all less than k in absolute value.
Let G be a bipartite graph with adjacency matrix A.(a) Show that A is not primitive.(b) Show that if l is an eigenvalue of A, so is -λ.
Compute the steady state growth rate of the population with the Leslie matrix L from the given exercise. Then use Exercise 18 to help find the corresponding distribution of the age classes.Exercise 19 in Section 3.7Data From Exercise 19 Section 3.7 1/2 1/4] 1/2 3/4.
Let A and B be similar matrices. Prove that the algebraic multiplicities of the eigenvalues of A and B are the same.
If A and B are invertible matrices, show that AB and BA are similar.
Prove Theorem 4.22(e).
Prove Theorem 4.22(c).a. det A = det Bb. A is invertible if and only if B is invertible.c. A and B have the same rank.
Use the method of Example 4.29 to compute the indicated power of the matrix. 12002 -1 0 -1 _0
Use the method of Example 4.29 to compute the indicated power of the matrix. ] 10 -1
Determine whether A is diagonalizable and, if so, find an invertible matrix P and a diagonal matrix D such that P-1AP = D. 1 1 1 1 1 1 0 2 A = %3D 0 0 2 1 L0 0 0 1
Determine whether A is diagonalizable and, if so, find an invertible matrix P and a diagonal matrix D such that P-1AP = D. 2 0 A = | 0
Determine whether A is diagonalizable and, if so, find an invertible matrix P and a diagonal matrix D such that P-1AP = D. A = 2
Show that A and B are not similar matrices. 1 0 3 1 2 2 2 , B = 4
Compute the determinants using cofactor expansion along any row or column that seems convenient. |0 a 0| |0
Compute(a) The characteristic polynomial of A,(b) The eigenvalues of A,(c) A basis for each eigenspace of A,(d) The algebraic and geometric multiplicity of each eigenvalue. 1 4 1 A = -1 2 0 ||
Compute(a) The characteristic polynomial of A,(b) The eigenvalues of A,(c) A basis for each eigenspace of A,(d) The algebraic and geometric multiplicity of each eigenvalue. 2 1 1 1 A = L0 0 0 2 3. 3.
Compute(a) The characteristic polynomial of A,(b) The eigenvalues of A,(c) A basis for each eigenspace of A,(d) The algebraic and geometric multiplicity of each eigenvalue. 3 1 0 0 A = 0 0 1 1
Compute(a) The characteristic polynomial of A,(b) The eigenvalues of A,(c) A basis for each eigenspace of A,(d) The algebraic and geometric multiplicity of each eigenvalue. 2 0 A = | -1 -1
Use Theorem 4.6 to find all values of k for which A is invertible. k k 0 A = | K 4 K k k
Prove Theorem 4.3(a).a. If A has a zero row (column), then det A = 0.
Find the determinants assuming that c| f = 4 a | 2d — 3g 2e — 3h 2f — 3і - зi h
Find the determinants assuming that c| f = 4 a + g b+h_ c + i| i
Use properties of determinants to evaluate the given determinant by inspection. Explain your reasoning. -3 0
Use the method of Example 4.5 to find all of the eigenvalues of the matrix A. Give bases for each of the corresponding eigenspaces. Illustrate the eigenspaces and the effect of multiplying eigenvectors by A as in Figure 4.8. A = 6. y 4 - Ax = 4x Ay = 2y У х -4 -3 -2 -1 1 2 3 -1+ -2+ -3+ -4- 3. 2.
The unit vectors x in R2and their images Ax under the action of a 2 Ã— 2 matrix A are drawn head-totail, as in Figure 4.7. Estimate the eigenvectors and eigenvalues of A from each €œeigenpicture.€Figure 4.7 y 2 х -2 2 -2+ -4 -4
Show that l is an eigenvalue of A and find one eigenvector corresponding to this eigenvalue. = 4
If a square matrix A has two equal rows, why must A have 0 as one of its eigenvalues?
If A3 = A, what are the possible eigenvalues of A?
Let Find all values of k for which:(a) A has eigenvalues 3 and -1.(b) A has an eigenvalue with algebraic multiplicity 2.(c) A has no real eigenvalues. k 2
IfFind 3, h i || за 2е — 4f f |За 2b — 4с с 3g 2h – 4i i
Let(a) Compute det A by cofactor expansion along any row or column.(b) Compute det A by first reducing A to triangular form. 3 5 A = | 3 5 9 11 [7
Mark each of the following statements true or false:(a) For all square matrices A, det(-A) = -det A.(b) If A and B are n × n matrices, then det(AB) = det (BA).(c) If A and B are n × n matrices whose columns are the same but in different orders, then det B det A.(d) If A is invertible, then
find an invertible matrix P and a matrix C of the form such that A = PCP -1.Sketch the first six points of the trajectory for the dynamical system and classify the origin as a spiral attractor, spiral repeller, or orbital center. a Ax with xo Xk+1 ||
find an invertible matrix P and a matrix C of the form such that A = PCP -1.Sketch the first six points of the trajectory for the dynamical system and classify the origin as a spiral attractor, spiral repeller, or orbital center. a Ax with xo Xk+1 ||
Find an invertible matrix P and a matrix C of the form such that A = PCP -1.Sketch the first six points of the trajectory for the dynamical system and classify the origin as a spiral attractor, spiral repeller, or orbital center. a Ax with xo Xk+1 ||
find an invertible matrix P and a matrix C of the form such that A = PCP -1.Sketch the first six points of the trajectory for the dynamical system and classify the origin as a spiral attractor, spiral repeller, or orbital center. a Ax with xo Xk+1 ||
the given matrix is of the formIn each case, A can be factored as the product of a scaling matrix and a rotation matrix. Find the scaling factor r and the angle of rotation. Sketch the first four points of the trajectory for the dynamical systemand classify the origin as a spiral attractor, spiral
the given matrix is of the formIn each case, A can be factored as the product of a scaling matrix and a rotation matrix. Find the scaling factor r and the angle of rotation. Sketch the first four points of the trajectory for the dynamical systemand classify the origin as a spiral attractor, spiral
the given matrix is of the formIn each case, A can be factored as the product of a scaling matrix and a rotation matrix. Find the scaling factor r and the angle of rotation. Sketch the first four points of the trajectory for the dynamical systemand classify the origin as a spiral attractor, spiral
the given matrix is of the formIn each case, A can be factored as the product of a scaling matrix and a rotation matrix. Find the scaling factor r and the angle of rotation. Sketch the first four points of the trajectory for the dynamical systemand classify the origin as a spiral attractor, spiral
Consider the dynamical systemxk + 1 = Axk.(a) Compute and plot x0,x1,x2,x3, for x0(b) Compute and plot x0,x1,x2,x3, for x0 (c) Using eigenvalues and eigenvectors, classify the origin as an attractor, repeller, saddle point, or none of these.(d) Sketch several typical trajectories of the
Consider the dynamical systemxk + 1 = Axk.(a) Compute and plot x0,x1,x2,x3, for x0(b) Compute and plot x0,x1,x2,x3, for x0 (c) Using eigenvalues and eigenvectors, classify the origin as an attractor, repeller, saddle point, or none of these.(d) Sketch several typical trajectories of the
Consider the dynamical systemxk + 1 = Axk.(a) Compute and plot x0,x1,x2,x3, for x0(b) Compute and plot x0,x1,x2,x3, for x0 (c) Using eigenvalues and eigenvectors, classify the origin as an attractor, repeller, saddle point, or none of these.(d) Sketch several typical trajectories of the
Consider the dynamical systemxk + 1 = Axk.(a) Compute and plot x0,x1,x2,x3, for x0(b) Compute and plot x0,x1,x2,x3, for x0 (c) Using eigenvalues and eigenvectors, classify the origin as an attractor, repeller, saddle point, or none of these.(d) Sketch several typical trajectories of the
Consider the dynamical systemxk + 1 = Axk.(a) Compute and plot x0,x1,x2,x3, for x0(b) Compute and plot x0,x1,x2,x3, for x0 (c) Using eigenvalues and eigenvectors, classify the origin as an attractor, repeller, saddle point, or none of these.(d) Sketch several typical trajectories of the
Consider the dynamical systemxk + 1 = Axk.(a) Compute and plot x0,x1,x2,x3, for x0(b) Compute and plot x0,x1,x2,x3, for x0 (c) Using eigenvalues and eigenvectors, classify the origin as an attractor, repeller, saddle point, or none of these.(d) Sketch several typical trajectories of the
Consider the dynamical systemxk + 1 = Axk.(a) Compute and plot x0,x1,x2,x3, for x0(b) Compute and plot x0,x1,x2,x3, for x0 (c) Using eigenvalues and eigenvectors, classify the origin as an attractor, repeller, saddle point, or none of these.(d) Sketch several typical trajectories of the
Consider the dynamical systemxk + 1 = Axk.(a) Compute and plot x0,x1,x2,x3, for x0(b) Compute and plot x0,x1,x2,x3, for x0 (c) Using eigenvalues and eigenvectors, classify the origin as an attractor, repeller, saddle point, or none of these.(d) Sketch several typical trajectories of the
Solve the system of differential equations in the given exercise using Theorem 4.41.Exercise 64Data From Exercise 64x′ = x + 3z, x(0) = 2y′ = x - 2y + z, y(0) = 3z′ = 3x + z, z(0) = 4

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