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Elementary Linear Algebra with Applications 9th edition Howard Anton, Chris Rorres - Solutions

(a) Prove that if A is a square matrix, then A and AT have the same eigenvalues (b) Show that A and AT need not have the same eigenspaces.
Suppose that the characteristic polynomial of some matrix, 4 is found to be p(λ) = (λ - 1)(λ - 3)2(λ-4)3 In each part, answer the question and explain your reasoning (a) What is the size o f A? (b) Is A invertible? (c) How many eigenspaces does A have?
Find bases for the eigenspaces of the matrices in Exercise 1.Exercise 1(a)(b)
Find the characteristic equations of the following matrices:
Find bases for the eigenspaces of the matrices in Exercise 7.Exercise 7Find the characteristic equations of the following matrices:
Let A be a 6 × 6 matrix with characteristic equation λ (λ - 1) (λ - 2)3 = 0. What are the possible dimensions for eigenspaces of A?
Find the geometric and algebraic multiplicity of each eigenvalue, and determine whether, A is diagonalizable. If so, find a matrix P that diagonalizes A, and determine P-1AP.1.2.
Find An if n is a positive integer and
Indicate whether each statement is always true or sometimes false. Justify your answer by giving a logical argument or a counterexample. (a) A square matrix with linearly independent column vectors is diagonalizable. (b) If A is diagonalizable, then there is a unique matrix P such that p-1AP is a
If A1, A2, ˆ™ ˆ™ ˆ™, Ak, ˆ™ ˆ™ ˆ™ is an infinite sequence of n Ã— n matrices, then the sequence is said to converge to the n Ã— n matrix , A if the entries in the ith row and jth column of the sequence converge to
Show that the Jordan block matrix Jn has λ = 1 as its only eigenvalue and that the corresponding eigenspace is span {e1}.
Use the method of Exercise 2 to determine whether the matrix is diagonalizable.
Find the characteristic equation of the given symmetric matrix, and then by inspection determine the dimensions of the eigenspaces.a.b. c.
Use the result in Exercise 17 of Section 7.1 to prove Theorem 7.3.2a for 2 × 2 symmetric.
Does there exist a 3 Ã— 3 symmetric matrix with eigenvalues Î»1 = -1, Î»2 = 3, Î»3 = 7 and corresponding eigenvectorsIf so, find such a matrix; if not, explain why not.
Show that if 0has no eigenvalues and consequently no eigenvectors.
Find the eigenvalues of the matrix
Let A be a square matrix such that A3 = A. What can you say about the eigenvalues of A?
(a) Show that if D is a diagonal matrix with nonnegative entries on the main diagonal, then there is a matrix S such that S2 = D.(b) Show that if A is a diagonalizable matrix with nonnegative eigenvalues, then there is a matrix S such that S2 = A.(c) Find a matrix S such that S2 = A, if
Prove: If A is a square matrix and p (λ) = det(λI - A) is the characteristic polynomial of A, then the coefficient of λn-1 in p (A) is the negative of the trace of A.
In advanced linear algebra, one proves the Cayley-Hamilton Theorem, which states that a square matrix A satisfies its 7. characteristic equation; that is, ifc0 + c1Î» + c2Î»2 + ˆ™ ˆ™ ˆ™ + cn-1Î»n-1 + Î»n = 0is the
Find the domain and codomain of T2 ○ T1, and find (T2 ○ T1)(x, y) (a) T1(x, y) = (2x, 3y), T2(x, y) = (x - y, x + y) (b) T1(x, y) = (x - y, y + z, x - z), T2(x, y, z) = (0, x + y + z)
Let T1: M22 †’ R and T2: M22 †’ M22 be the linear transformations given by T1 (A) = tr(A) and T2(A) = AT.(a) Find (T1 o T2) (A), where(b) Can you find (T2 o T1)(A)? Explain.
Let T: R3 → R3 be the orthogonal projection of R3 onto the xy-plane. Show that T o T = T.
T: V → R, where V an inner product space, and T(u) = ||u||. Determine whether the function is a linear transformation. Justify your answer.
LetD(f) = f€²(x)andbe the linear transformations in Examples Example 11 and Example 12. Find (J o D)(f) for (a) f(x) = sin x (b) f(x) = ex + 3
Indicate whether each statement is always true or sometimes false. Justify your answer by giving a logical argument or a counterexample. In each part, V and W are vector spaces. (a) If T(c1v1 + c2v2) = c1T(v1) + c2T(v2) for vectors v1 and v2 in Land all scalars c1 and c2, then T is a linear
Refer to Section 4.4. Are the transformations from Pn to Pm that correspond to linear transformations from Rn+1 to Rm+1 necessarily linear transformation from Pn to Pm?
T: M22 → M23, where B is a fixed 2 × 3 matrix and T(A) = AB. Determine whether the function is a linear transformation. Justify your answer.
Let T: R2 → R2 be the linear operator given by the formula T(x, y) = (2x - y, -8x + 4y) Which of the following vectors are in R(T)? (a) (1, -4) (b) (5, 0)
Let T: R3 → V be a linear transformation from R3 to any vector space. Show that the kernel of T is a line through the origin, a plane through the origin, the origin only, or all of R3.
Let T: R3 †’ R3 be multiplication by(a) Show that the kernel of T is a line through the origin, and find parametric equations for it. (b) Show that the range of T is a plane through the origin, and find an equation for it.
For the positive integer n > 1, let T: Mnn → R be the linear transformation defined by T(A) = tr(A), for A an n × n matrix with real entries. Determine the dimension of ker(T).
Let D: V → W be the differentiation transformation D(f) = f′(x), where V = C3 (-∞, ∞) and W = F(-∞, ∞). Describe the kernels of D o D and D o D o D.
(a) If T: R3 → R3 is a linear operator, and if the kernel of T is a line through the origin, then what kind of geometric object is the range of T? Explain your reasoning. (b) If T: R3 → R3 is a linear operator, and if the range of T is a plane through the origin, then what kind of geometric
Let T: R4 → R3 be the linear transformation given by the formula T(x1, x2, x3, x4) = (4x1 + x2 - 2x3 - 3x4, 2x1 + x2 + x3 - 4x4, 6x1 - 9x3 + 9x4) Which of the following are in R(T)? (1, 3, 0)
Find a basis for the kernel of (a) The linear operator in Exercise 1 (b) The linear transformation in Exercise 5.
In each part, find ker(T), and determine whether the linear transformation T is one-to-one. (a) T: R2 → R2, where T(x, y) = (y, x) (b) T: R2 → R2, where T(x, y) = (x + y, x - y) (c) T: R2 → R3, where T(x, y) = (x - y, y - x, 2x - 2y)
Let T1: P2 → P3 and T2: P3 → P3 be the linear transformations given by the formulas T1(p(x)) = xp(x) and T2(p(x)) = p(x + 1) Find formulas for T1-l (p(x)), T2-1 (p(x)), and (T2 o T1)-1 (p(x))
In each part, determine whether the linear operator T: M22 †’ M22 is one-to-one. If so, find(a) (b)
Prove that if T: V → W is a one-to-one linear transformation, then T-1 → V is a linear transformation.
Let a be a fixed vector in R3. Does the formula T(v) = a × v define a one-to-one linear operator on R3? Explain your reasoning.
In each part, let T: R3 †’ R3 be multiplication by A. Determine whether T has an inverse; if so, find(a) (b)
As indicated in the accompanying figure, let T: R2 †’ R2 be the orthogonal projection on the line y = x(a) Find the kernel of T.(b) Is T one-to-one? Justify your conclusion.
Letbe the matrix of T: P2 †’ P2 with respect to the basis B = (v1, v2, v3), where v1 = 3x + 3x2, v2 = -1 + 3x + 2x2, v3 = 3 + 7x + 2x2. (a) Find [T(v1)]B, [T(v2)]B, and [T(v3)]B. (b) Find T(v1), T(v2), and T(v3). (c) Find a formula for T(a0 + a1x + a2x2). (d) Use the formula obtained in
Let T1: P1 → P2 be the linear transformation defined by T1(c0 + c1x) = 2c0 - 3c1x and let T2: P2 → P3 be the linear transformation defined by T2(c0 + c1x + c2x2) = 3c0x + 3c1x2 + 3c2x3 Let B = (1, x), B′′ = {1, x, x2}, and Bt = {1, x, x2, x3}. (a) Find [T2 o T1]B′,B, [T2] B′,B′′,
Show that if T: V → V is a contraction or a dilation of V(Example 4 of Section 8.1), then the matrix of T with respect to any basis for V is a positive scalar multiple of the identity matrix.
Prove that if B and B′ are the standard bases for Rn and Rm, respectively, then the matrix for a linear transformation T: Rn → Rm with respect to the bases B and B′ is the standard matrix for T.
In each part, B = {f1, f2, f3} is a basis for a subspace V of the vector space of real-valued functions defined on the real line. Find the matrix with respect to B of the differentiation operator D: V → V. f1 = e2x, f2 = xe2x, f3 = x2e2x
LetAnd let be the matrix for T: R2 †’ R2 with respect to the basis B = (v1, v2). (a) Find [T(v1)]B and [T(v2)]B. (b) Find T(v1) and T(v2). (c) Find a formula for (d) Use the formula obtained in (c) to compute
Find the matrix of T with respect to the basis B, and use Theorem 8.5.2 to compute the matrix of T with respect to the basis B€².T: R2 †’ R2 is defined byB= {u1, u2} and B€² = {v1, v2}, where
In each part, find a basis for R2 relative to which the matrix for T is diagonal.
Let T: P2 → P2 be define by T(a0 + a1x + a2x2) = (5a0 + 6a1 + 2a2) - (a1 + 8a2)x + (a0 - 2a2)x2 (a) Find the eigenvalues of T. (b) Find bases for the eigenspaces of T.
Let λ be an eigenvalue of a linear operator T: V → V. Prove that the eigenvectors of T corresponding to λ are the nonzero vectors in the kernel of λI - T.
Let C and D be m × n matrices, and let B = (v1, v2, ..., vn) be a basis for a vector V. Show that if C[x]B = D[x]B for all x in V, then C = D.
Indicate whether each statement is always true or sometimes false. Justify your answer by giving a logical argument or a counterexample. (a) A matrix cannot be similar to itself. (b) If A is similar to B, and B is similar to C, then A is similar to C. (c) If A and B are similar and B is singular,
Prove that the trace is a similarity invariant.
T: R2 †’ R2 is the rotation about the origin through 45°; B and B€² are the bases in Exercise 1.Find the matrix of T with respect to the basis B, and use Theorem 8.5.2 to compute the matrix of T with respect to the basis B€².Exercise 1T: R2 †’ R2 is
T: R3 †’ R3 is the orthogonal projection on the x,y-plane; B and B€² are as in Exercise 4.Find the matrix of T with respect to the basis B, and use Theorem 8.5.2 to compute the matrix of T with respect to the basis B€².Exercise 4T: R3 †’ R3 is defined byB is
T: P1 → P1 is defined by T(a0 + a1x) = a0 + a1{x + 1); B = {p1, p2} and B′ = {q1, q2}, where p1 = 6 + 3x, p2 = 10 + 2x, q1 = 2, q2 = 3 + 2x. Find the matrix of T with respect to the basis B, and use Theorem 8.5.2 to compute the matrix of T with respect to the basis B′.
Prove that the following is similarity invariant: Rank
Which of the transformations in Exercise 1 of Section 8.3 are onto? Exercise 1 of Section 8.3 In each part, find ker(T), and determine whether the linear transformation T is one-to-one. (a) T: R2 → R2, where T(x, y) = (y, x) (b) T: R2 → R2, where T(x, y) = (0, 2x + 3y) (c) T: R2 → R2, where
How could differentiation of functions in the vector space span {1, sin(x), cos(x), sin(2x), cos(2x)} be computed by matrix multiplication in R5? Use your method to find the derivative of 3 - 4 sin(x) + sin(2x) + 5cos(2x).
Which of the transformations in Exercise 3 of Section 8.3 are onto?Exercise 3 of Section 8.3In each part, let T: R3 †’ R3 be multiplication by A. Determine whether T has an inverse; if so, find(a) (b) (c) (d)
Which of the following transformations are bijections? (a) T: P2(x) → P3(x), T(P(x)) = xp(x) (b) T: M22 → M22, T(A) = AT (c) T: R4 → R3, T(x, y, z, w) = (x, y, 0) (d) T: P3 → R3, T(a + bx + cx2 + dx3) = (b , c , d)
Prove: There can be a surjective linear transformation from V to W only if dim(v) ≥ dim (W).
Let S be the standard basis for Rn. Prove Theorem 8.6.2 by showing that the linear transformation T: V → Rn that maps v ∈ V to its coordinate vector (v)s in Rn is an isomorphism.
Let T: M22 †’ M22 be the linear operator defined byFind the rank and nullity of T.
Let L: M22 → M22 be the linear operator defined by L(M) = MT. Find the matrix for L with respect to the standard basis for M22.
Let B = {u1, u2, u3} be a basis for a vector space V, and let T: V †’ V be a linear operator such thatFind [T]B€², where B€² = {v1, v2, v3}, is the basis for V defined by v1 = u1, v2 = u1 + u2, v3 = u1 + u2 + u3
Suppose that T: V †’ V is a linear operator and B is a basis for V such that for any vector x in V,Find [T]B.
(a) Show that if f = f (x), then the function D: C2(-∞, ∞) → F(-∞, ∞) defined by D(f) = f′′ (x) is a linear transformation. (b) Find a basis for the kernel of D. (c) Show that the functions satisfying the equation D(f) = f(x) form a two-dimensional subspace of C2(-∞, ∞), and find
Let x1, x2, and x3 be distinct real numbers such that x1(a) Show that T is a linear transformation. (b) Show that T is one-to-one. (c) Verify that if a1, a2, and a3 are any real numbers, then Where (d) What relationship exists between the graphs of the function a1P1(x) + a2P2(x) + a3P3(x) and the
Let D: Pn †’ Pn be the differentiation operator D(p) = p€². Show that the matrix for D with respect to the basis B = {l, x, x2,..., xn} is
Let J: Pn †’ Pn+1 be the integration transformation defined by
Let v0 be a fixed vector in an inner product space V, and let T: V → V be defined by T(v) = (v, v0)v0. Show that T is a linear operator on V.
Let {e1, e2, e3, e4} be the standard basis for R4, and let T: R4 → R3 be the linear transformation for which T(e1) = (1, 2, 1), T(e2) = (0, 1, 0), T(e3) = (1, 3, 0), T(e4) = (1, 1, 1), Find bases for the range and kernel of T.
Let B = {v1, v2, v3, v4} be a basis for a vector space L, and let T: V → V be the linear operator for which T(v1) = v1 + v2 + v3 + 3v4 T(v2) = v1 - v2 + 2v3 + 2v4 T(v3) = 2v1 - 4v2 + 5v3 + 3v4 T(v4) = -2v1 + 6v2 - 6v3 - 2v4 (a) Find the rank and nullity of T. (b) Determine whether T is one-to-one.
Let A and B be similar matrices. Prove: AT and BT are similar.
(a) Solve the system y′1 = y1 + 4y2 y′2 = 2y1 + 3y2 (b) Find the solution that satisfies the initial conditions y1(0) = 0, y2(0) = 0.
Consider the system of differential equations y′ = Ay where A is a 2 × 2 matrix. For what values of a11, a12, a21, a22 do the component solutions y1(t), y2(t) tend to zero as t → ∞? In particular, what must be true about the determinant and the trace of A for this to happen?
Use diagonalization to solve the system y′1 = 2y1 + y2 + t, y′2 = y1 + 2y2 + 2t by first writing it in the form y′ = Ay + f. Note the presence of a forcing function in each equation.
(a) Solve the system y′1 = 4y1 + y3 y′2 = -2y1 + y2 y′3 = -2y1 + y3 (b) Find the solution that satisfies the initial conditions y1(0) = -1, y2(0) = 1, y3(0) = 0.
Show that every solution of y′ = ay has the form y = ceax.
It is possible to solve a single differential equation by expressing the equation as a system and then using the method of this section. For the differential equation y′′ - y′ - 6y = 0, show that the substitutions y1 = y and y2 = y′ lead to the system y′1 = y2 y′2 = 6y1 + y2 Solve this
Discuss: How can the procedure in Exercise 7 be used to solve y‴ - 6y″ + 11y′ - 6y = 0? Carry out your ideas. Exercise 7 It is possible to solve a single differential equation by expressing the equation as a system and then using the method of this section. For the differential equation
Find the standard matrix for the linear operator T: R2 †’ R2 that maps a point (x, 7) into (see the accompanying figure)(a) Its reflection about the line y = -x(b) Its reflection through the origin(c) Its orthogonal projection on the x-axis(d) Its orthogonal projection on the y-axis
In each part, find a single matrix that performs the indicated succession of operations: (a) Compresses by a factor of 1/2 in the x-direction, then expands by a factor of 5 in the y-direction (b) Reflects about y = x, then rotates through an angle of 180° about the origin
By matrix inversion, show the following: (a) The inverse transformation for a compression along an axis is an expansion along that axis. (b) The inverse transformation for a reflection about a coordinate axis is a reflection about that axis.
In parts, find the equation of the image of the line y = 2x under (a) A shear of factor 3 in the x-direction (b) A reflection about y = x (c) A rotation of 60° about the origin
Use the Exercise 20 to prove parts (b) and (c) of Theorem 2.A line in the plane has an equation of the form Ax + By + C = 0, where A and B are not both zero. Use the method of Example 8 to show that the image of this line under multiplication by the invertible matrixhas the equation A€²x
In R3 the shear in the xy-direction with factor k is the linear transformation that moves each point (x, y, z) parallel to the x-plane to the new position (x + ky, y + kz, z)- (See the accompanying figure.)Find the standard matrix for the shear in the xy-direction with factor k.
Find the standard matrix for the linear operator T: R3 → R3 that maps a point (x, y, z) into Its reflection through the xz-plane
Find the standard matrix for the linear operator T: R3 → R3 that (a) Rotates each vector 90° counterclockwise about the z-axis (looking along the positive z axis toward the origin) (b) Rotates each vector 90° counterclockwise about the y-axis (looking along the positive j axis toward the origin)
Find the least squares straight line fit to the three points (0, 0), (1, 2), and (2, 7).
Find the quadratic polynomial that best fits the four points (2, 0), (3, -10), (5, -48), and (6, -76).
Show that the matrix M in Equation 2 has linearly independent columns if and only if at least two of the numbers x1, x2, ..., xn are distinct.
Find the least squares approximation of f(x) = 1 + x over the interval [0, 2π] by A trigonometric polynomial of order 2 or less
(a) Find the least squares approximation of x over the interval [0, 1] by a function of the form a + bex. (b) Find the mean square error of the approximation.
(a) Find the least squares approximation of sin πx over the interval [-1, 1] by a polynomial of the form a0 + a1x + a2x2. (b) Find the mean square error of the approximation.

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