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All Textbook Solutions for Elementary Linear Algebra (MindTap Course List)
48CR49CR50CRFinding the Inverse of a Linear Transformation In Exercise 49-52, determine whether the linear transformation is invertible. If it is, find its inverse. T:R2R2, T(x,y)=(x,y)Finding the Inverse of a Linear Transformation In Exercise 49-52, determine whether the linear transformation is invertible. If it is, find its inverse. T:R3R2, T(x,y,z)=(x+y,yz)One-to-One, Onto, and Invertible Transformations In Exercises 53-56, determine whether the linear transformation represented by the matrix A is a one-to-one, b onto, and c invertible. A=[6001]One-to-One, Onto, and Invertible Transformations In Exercises 53-56, determine whether the linear transformation represented by the matrix A is a one-to-one, b onto, and c invertible. A=[11401]One-to-One, Onto, and Invertible Transformations In Exercises 53-56, determine whether the linear transformation represented by the matrix A is a one-to-one, b onto, and c invertible. A=[111011]One-to-One, Onto, and Invertible Transformations In Exercises 53-56, determine whether the linear transformation represented by the matrix A is a one-to-one, b onto, and c invertible. A=[407551002]Finding the Image Two Ways InExercises 57 and 58, find T(v)by using a the standard and b the matrix relative to B and B. T:R3R3 T(x,y)=(x,y,x+y),v=(0,1)B={(1,1),(1,1)},B={(0,1,0),(0,0,1),(1,0,0)}Finding the Image Two Ways In Exercises 57 and 58, find T(v)by using a the standard and b the matrix relative to B and B. T:R2R2 T(x,y)=(2y,0),v=(1,3)B={(2,1),(1,0)},B={(1,0),(2,2)}Finding a Matrix for a Linear Transformation In Exercises 59 and 60, find the matrix A for T relative to the basis B. T:R2R2,T(x,y)=(x3y,yx) B={(1,1),(1,1)}60CR61CR62CR63CR64CR65CR66CRSum of Two Linear Transformations In Exercises 67 and 68, consider the sum S+T of two linear transformations S:VW and T:VW, defined as (S+T)(v)=S(v)+T(v). Proof Prove that rank(S+T)rank(S)+rank(T).68CR69CR70CRLet V be an inner product space. For a fixed nonzero vector v0 in V, let T:VR be the linear transformation T(v)=v,v0. Find the kernel, range, rank, and nullity of T.Calculus Let B={1,x,sinx,cosx} be a basis for a subspace W of the space of continuous functions and Dx be the differential operator on W. Find the matrix for Dx relative to the basis B. Find the range and kernel of Dx.73CR74CR75CR76CR77CR78CR79CR80CR81CR82CR83CR84CR85CR86CR87CR88CR89CR90CR91CR92CR93CR94CR95CR96CR97CR98CRTrue or False? In Exercises 99-102, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. a Reflection that map a point in the xy-plane to its mirror image across the line y=x are linear transformations that are defined by the matrix [1001]. b Horizontal expansions or contractions are linear transformations that are defined by the matrix [k001].True or False? In Exercises 99-102, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. a Reflection that map a point in the xy-plane to its mirror image across the x-axis are linear transformations that are defined by the matrix [1001]. b Vertical expansions or contractions are linear transformations that are defined by the matrix [100k].101CR102CRVerifying Eigenvalues and Eigenvectors in Exercises 1-6, verify that iis an eigenvalue of A and that xiis a corresponding eigenvector. A=[2002], 1=2,x1=(1,0)2=2,x2=(0,1)Verifying Eigenvalues and EigenvectorsIn Exercises 1-6, verify that i is an eigenvalues of A and that Xi is a corresponding eigenvector. A=[4523], 1=1,X1=(1,1)2=2,X2=(5,2)Verifying Eigenvalues and EigenvectorsIn Exercises 1-6, verify that i is an eigenvalues of A and that Xi is a corresponding eigenvector. A=[231012003], 1=2,X1=(1,0,0)2=1,X2=(1,1,0)3=3,X3=(5,1,2)Verifying Eigenvalues and Eigenvectors in Exercises 1-6, verify that i is an eigenvalues of A and that Xi is a corresponding eigenvector. A=[223216120], 1=5,X1=(1,2,1)2=3,X2=(2,1,0)3=3,X3=(3,0,1)Verifying Eigenvalues and EigenvectorsIn Exercises 1-6, verify that i is an eigenvalues of A and that Xi is a corresponding eigenvector. A=[010001100], 1=1,X1=(1,1,1)Verifying Eigenvalues and EigenvectorsIn Exercises 1-6, verify that i is an eigenvalues of A and that Xi is a corresponding eigenvector. A=[413021003], 1=4,X1=(1,0,0)2=2,X2=(1,2,0)3=3,X3=(2,1,1)7E8EDetermining Eigenvectors In Exercise 9-12, determine whether X is an eigenvector of A. A=[7224] a X=(1,2) b X=(2,1) c X=(1,2) d X=(1,0)Determining Eigenvectors In Exercise 9-12, determine whether X is an eigenvector of A. A=[31052] a X=(4,4) b X=(8,4) c X=(4,8) d X=(5,3)Determining Eigenvectors In Exercise 9-12, determine whether X is an eigenvector of A. A=[111202331] a X=(2,4,6) b X=(2,0,6) c X=(2,2,0) d X=(1,0,1)12E13E14ECharacteristic Equation, Eigenvalues, and Eigenvectors In Exercise 15-28, find a the characteristics equation and b the eigenvalues and corresponding eigenvectors of the matrix. [6321]Characteristic Equation, Eigenvalues, and Eigenvectors in Exercise 15-28, find a the characteristics equation and b the eigenvalues and corresponding eigenvectors of the matrix. [1428]Characteristic Equation, Eigenvalues, and Eigenvectors In Exercise 15-28, find a the characteristic equation and b the eigenvalues and corresponding eigenvectors of the matrix. [1221]18ECharacteristic Equation, Eigenvalues, and EigenvectorsIn Exercise 15-28, find a the characteristics equation and b the eigenvalues and corresponding eigenvectors of the matrix. [132121]20ECharacteristic Equation, Eigenvalues and Eigenvectors In Exercise 15-28, find a the characteristic equation and b the eigenvalues and corresponding eigenvectors of the matrix. [223032012]Characteristic Equation, Eigenvalues and Eigenvector, In Exercise 15-28, find a the characteristic equation and b the eigenvalues and corresponding eigenvectors of the matrix. [321002020]Characteristic Equation, Eigenvalues and Eigenvector, In Exercise 15-28, find a the characteristic equation and b the eigenvalues and corresponding eigenvectors of the matrix. [122252663]24ECharacteristic Equation, Eigenvalues and Eigenvector, In Exercise 15-28, find a the characteristic equation and b the eigenvalues and corresponding eigenvectors of the matrix. [0354410004]Characteristic Equation, Eigenvalues and Eigenvector, In Exercise 15-28, find a the characteristic equation and b the eigenvalues and corresponding eigenvectors of the matrix. [1325221321032928]Characteristic Equation, Eigenvalues and Eigenvector, In Exercise 15-28, find a the characteristic equation and b the eigenvalues and corresponding eigenvectors of the matrix. [2000020000310040]Characteristic Equation, Eigenvalues and Eigenvector, In Exercise 15-18, find a the characteristic equation and b the eigenvalues and corresponding eigenvectors of the matrix. [5000140000130004]29E30E31E32E33E34E35E36E37E38E39EFinding EigenvaluesIn Exercises 29-40, use a software program or a graphing utility to find the eigenvalues of the matrix. [1333143320111000]Eigenvalues of Triangular and Diagonal Matrices In Exercises 41-44, find the eigenvalues of the triangular or diagonal matrix. [201034001]Eigenvalues of Triangular and Diagonal Matrices In Exercises 41-44, find the eigenvalues of the triangular or diagonal matrix. [500370423]43EEigenvalues of Triangular and Diagonal Matrices In Exercises 41-44, find the eigenvalues of the triangular or diagonal matrix. [1200005400000000034]Eigenvalues and Eigenvectors of Linear TransformationsIn Exercises 45-48, consider the linear transformation T:RnRnwhose matrix Arelative to the standard basis is given. Find a the eigenvalues of A, b a basis for each of the corresponding eigenspaces, and c the matrix Afor Trelative to the basis B, where Bis made up of the basis vectors found in part b. [2215]46EEigenvalues and Eigenvectors of Linear TransformationsIn Exercises 45-48, consider the linear transformation T:RnRnwhose matrix Arelative to the standard basis is given. Find a the eigenvalues of A, b a basis for each of the corresponding eigenspaces, and c the matrix Afor Trelative to the basis B, where Bis made up of the basis vectors found in part b. [021131001]Eigenvalues and Eigenvectors of Linear TransformationsIn Exercises 45-48, consider the linear transformation T:RnRn whose matrix A relative to the standard basis is given. Find a the eigenvalues of A, b a basis for each of the corresponding eigenspaces, and c the matrix A fot T relative to the basis B, where B is made up of the basis vectors found in part b. [314240556]Cayley-Hamilton TheoremIn Exercises 49-52, demonstrate the Cayley-Hamilton Theorem for the matrix A. The Cayley-Hamilton Theorem states that a matrix satisfies its characteristic equation. For example, the characteristic equation of A=[1325]is, 26+11=0, and by the theorem you have, A26A+11I2=O. A=[5073]Cayley-Hamilton TheoremIn Exercises 49-52, demonstrate the Cayley-Hamilton Theorem for the matrix A. The Cayley-Hamilton Theorem states that a matrix satisfies its characteristic equation. For example, the characteristic equation of A=[1325]is, 26+11=0, and by the theorem you have, A26A+11I2=O. A=[6115]51E52E53E54E55E56E57EProof Prove that A and AT have the same eigenvalues. Are the eigenspaces the same?59EDefine T:R2R2 by T(v)=projuv Where u is a fixed vector in R2. Show that the eigenvalues of A the standard matrix of T are 0 and 1.61E62E63E64E65EShow that A=[0110] has no real eigenvalues.True or False? In Exercises 67 and 68, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. a The scalar is an eigenvalue of an nn matrix A when there exists a vector x such that Ax=x. b To find the eigenvalues of an nn matrix A. you can solve the characteristic equation det(IA)=0.True or False? In Exercises 67 and 68, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. a Geometrically, if is an eigenvalue of a matrix A and x is an eigenvector of A corresponding to , then multiplying x by A produce a vector x parallel to x. b If A is nn matrix with an eigenvalue , then the set of all eigenvectors of is a subspace of Rn.Finding the Dimension of an Eigenspace In Exercises 69-72, find the dimension of the eigenspace corresponding to the eigenvalue =3. A=[300030003]Finding the Dimension of an Eigenspace In Exercises 69-72, find the dimension of the eigenspace corresponding to the eigenvalue =3. A=[310030003]71E72E73E74E75EDefine T:P2P2 by T(a0+a1x+a2x2)=(2a0+a1a2)+(a1+2a2)xa2x2. Find the eigenvalues and the eigenvectors of T relative to the standard basis {1,x,x2}.77EFind all values of the angle for which the matrix A=[cossinsincos] has real eigenvalues. Interpret your answer geometrically.79E80E81EDiagonalizable Matrices and Eigenvalues In Exercise 1-6, a verify that A is diagonalizable by finding P1AP, and b use the result of part a and Theorem 7.4 to find the eigenvalues of A. A=[1136310],P=[3411]Diagonalizable Matrices and Eigenvalues In Exercise 1-6, a verify that A is diagonalizable by finding P1AP, and b use the result of part a and Theorem 7.4 to find the eigenvalues of A. A=[1315],P=[3111]Diagonalizable Matrices and Eigenvalues In Exercise 1-6, a verify that A is diagonalizable by finding P1AP, and b use the result of part a and Theorem 7.4 to find the eigenvalues of A. A=[3222],P=[1221]Diagonalizable Matrices and Eigenvalues In Exercise 1-6, a verify that A is diagonalizable by finding P1AP, and b use the result of part a and Theorem 7.4 to find the eigenvalues of A. A=[4523],P=[1512]Diagonalizable Matrices and Eigenvalues In Exercise 1-6, a verify that A is diagonalizable by finding P1AP, and b use the result of part a and Theorem 7.4 to find the eigenvalues of A. A=[110030425],P=[013040122]6EDiagonalizing a Matrix In Exercise 7-14, find if possible a nonsingular matrix P such that P1AP is diagonal. Verify thatP1APis a diagonal matrix with the eigenvalues on the main diagonal A=[6321] See Exercise 15, section 7.1. Characteristic Equation, Eigenvalues, and Eigenvectors in Exercise 15-28, find a the characteristics equation and b the eigenvalues and corresponding eigenvectors of the matrix. [6321]8EDiagonalizing a Matrix In Exercise 7-14, find if possible a nonsingular matrix P such that P1APis diagonal. Verify that P1APis a diagonal matrix with the eigenvalues on the main diagonal A=[223032012] See Exercise 20, section 7.1. Characteristic Equation, Eigenvalues, and Eigenvectors in Exercise 15-28, find a the characteristics equation and b the eigenvalues and corresponding eigenvectors of the matrix. [223032012]Diagonalizing a Matrix In Exercise 7-14, find if possible a nonsingular matrix P such that P1AP is diagonal. Verify that P1AP is a diagonal matrix with the eigenvalues on the main diagonal A=[321002020] See Exercise 22, section 7.1. Characteristic Equation, Eigenvalues, and Eigenvectors In Exercise 15-28, find a the characteristics equation and b the eigenvalues and corresponding eigenvectors of the matrix. [321002020]Diagonalizing a Matrix In Exercise 7-14, find if possible a nonsingular matrix P such that P1AP is diagonal. Verify that P1AP is a diagonal matrix with the eigenvalues on the main diagonal A=[122252663] See Exercise 23, section 7.1. Characteristic Equation, Eigenvalues, and EigenvectorsIn Exercise 15-28, find a the characteristics equation and b the eigenvalues and corresponding eigenvectors of the matrix. [122252663]Diagonalizing a Matrix In Exercise 7-14, find if possible a nonsingular matrix P such that P1AP is diagonal. Verify that P1AP is a diagonal matrix with the eigenvalues on the main diagonal A=[323349125] See Exercise 24, section 7.1. Characteristic Equation, Eigenvalues, and Eigenvectors In Exercise 15-28, find a the characteristics equation and b the eigenvalues and corresponding eigenvectors of the matrix. [323349125]Diagonalizing a Matrix In Exercise 7-14, find if possible a nonsingular matrix P such that P1AP is diagonal. Verify that P1AP is a diagonal matrix with the eigenvalues on the main diagonal A=[100121102]14EShow That a Matrix Is Not Diagonalizable In Exercise 15-22, show that the matrix is not diagonalizable. [0050]16EShow That a Matrix Is Not Diagonalizable In Exercise 15-22, show that the matrix is not diagonalizable. [7707]Show That a Matrix Is Not Diagonalizable In Exercise 15-22, show that the matrix is not diagonalizable. [1021]Show That a Matrix Is Not Diagonalizable In Exercise 15-22, show that the matrix is not diagonalizable. [121014002]Show That a Matrix Is Not Diagonalizable In Exercise 15-22, show that the matrix is not diagonalizable. [322023002]21E22EDetermine a Sufficient Condition for Diagonalization In Exercises 23-26, find the eigenvalues of the matrix and determine there is a sufficient number of eigenvalues to guarantee that the matrix is diagonalizable by Theorem 7.6. [1111]Determine a Sufficient Condition for Diagonalization In Exercises 23-26, find the eigenvalues of the matrix and determine there is a sufficient number of eigenvalues to guarantee that the matrix is diagonalizable by Theorem 7.6. [2052]Determine a Sufficient Condition for Diagonalization In Exercises 23-26, find the eigenvalues of the matrix and determine there is a sufficient number of eigenvalues to guarantee that the matrix is diagonalizable by Theorem 7.6. [323349125]Determine a Sufficient Condition for Diagonalization In Exercises 23-26, find the eigenvalues of the matrix and determine there is a sufficient number of eigenvalues to guarantee that the matrix is diagonalizable by Theorem 7.6. [432011002]Finding a Basis In Exercises 27-30, find a basis B for the domain of T such that the matrix for T relative to B is diagonal. T:R2R2:T(x,y)=(x+y,x+y)Finding a Basis In Exercises 27-30, find a basis B for the domain of T such that the matrix for T relative to B is diagonal. T:R3R3:T(x,y,z)=(2x+2y3z,2x+y6z,x2y)29E30E31E32E33EFinding a Power of a Matrix In Exercises 33-36, use the result of Exercise 31 to find the power of A shown. A=[1320],A7 Proof Let A be a diagonalizable nn matrix and let P be an invertible nn matrix such that B=P1AP is the diagonal form of A. Prove that Ak=PBkP1, where k is a positive integer.35E36ETrue or False? In Exercises 37 and 38, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. a If A and B are similar nn matrix, then they have always the same characteristics polynomial equation. b The fact that an nn matrix A has n distinct eigenvalues does not guarantee that A is diagonalizable.True or False? In Exercises 37 and 38, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. a If A is a diagonalizable matrix, then it has n linearly independent eigenvectors. b If an nn matrix A is diagonalizable, then it must have n distinct eigenvalues.Are the two matrices similar? If so, find a matrix P such that B=P1AP. A=[100020003]B=[300020001]40E41EProof Prove that if matrix A is diagonalizable, then AT is diagonalizable.Proof Prove that if matrix A is diagonalizable with n real eigenvalues 1,2,,n, then |A|=12n.44E45EGuide Proof Prove nonzero nilpotent matrices are not diagonalizable Getting started: From Exercises 80 in Section 7.1, you know that 0 is the only eigenvalue of the nilpotent matrix A. Show that it is impossible for A to be diagonalizable. i Assume A is diagonalizable, so there exists an invertible matrix P such that P1AP=D, here D is the zero matrix. ii Find A in term of P,P1 and D. iii Find a contradiction and conclude that nonzero nilpotent matrices are not diagonalizable.47ECAPSTONE Explain how to determine whether an nn matrix A is diagonalizable using a similar matrices, b eigenvectors, and c distinct eigenvalues.49EShowing That a Matrix Is Not Diagonalizable In Exercises 49 and 50, show that the matrix is not diagonalizable. [00k0],k0Determining Whether a Matrix Is Symmetric In Exercises 1 and 2, determine whether the matrix is symmetric. [421312121]2EProof In Exercise 3-6, prove that the symmetric matrix is diagonalizable. A=[00a0a0a00]4E5E6EFinding Eigenvalues and Dimensions of Eigen spaces In Exercise 7-18, find the eigenvalues of the symmetric matrix. For each eigenvalue, find the dimension of the corresponding eigenspace. [2112]Finding Eigenvalues and Dimensions of Eigen spaces In Exercises 7-18, find the eigenvalues of the symmetric matrix. For each eigenvalue, find the dimension of the corresponding eigenspace. [3003]Finding Eigenvalues and Dimensions of Eigen spaces In Exercises 7-18, find the eigenvalues of the symmetric matrix. For each eigenvalue, find the dimension of the corresponding eigenspace. [300020002]Finding Eigenvalues and Dimensions of Eigen spaces In Exercises 7-18, find the eigenvalues of the symmetric matrix. For each eigenvalue, find the dimension of the corresponding eigenspace. [211121112]Finding Eigenvalues and Dimensions of Eigen spaces In Exercises 7-18, find the eigenvalues of the symmetric matrix. For each eigenvalue, find the dimension of the corresponding eigenspace. [022202220]Finding Eigenvalues and Dimensions of Eigen spaces In Exercises 7-18, find the eigenvalues of the symmetric matrix. For each eigenvalue, find the dimension of the corresponding eigenspace. [044420402]Finding Eigenvalues and Dimensions of Eigen spaces In Exercises 7-18, find the eigenvalues of the symmetric matrix. For each eigenvalue, find the dimension of the corresponding eigenspace. [011101111]14E15E16E17E18EDetermine Whether a Matrix Is Orthogonal In Exercise 19-32, determine whether the matrix is orthogonal. If the matrix is orthogonal, then show that the column vectors of the matrix form an orthonormal set. [22222222]20E21E22E23E24E25E26E27E28E29E30E31E32E33E34E35EEigenvectors of Symmetric Matrix In Exercises 33-38, show that any two eigenvectors of the symmetric matrix corresponding to distinct eigenvalues are orthogonal. [300030002]37E38E39EOrthogonally Diagonalizable Matrices In Exercise 39-42, determine whether the matrix is orthogonally diagonalizable. [323212323]41E42E43E44EOrthogonal Diagonalization In Exercise 43-52, find a matrix P such that PTAP orthogonally diagonalizes A Verify that PTAP gives the correct diagonal form. A=[2221]Orthogonal Diagonalization In Exercise 43-52, find a matrix Psuch that PTAPorthogonally diagonalizesAVerify that PTAPgives the correct diagonal form. A=[011101110]Orthogonal Diagonalization In Exercise 4-52, find a matrix Psuch that PTAPorthogonally diagonalizes A. Verify that PTAPgives the correct diagonal form. A=[0101010501005]48E49EOrthogonal Diagonalization In Exercise 43-52, find a matrix Psuch that PTAPorthogonally diagonalizesA. Verify that PTAPgives the correct diagonal form. A=[224224444]Orthogonal Diagonalization In Exercise 4-52, find a matrix Psuch that PTAPorthogonally diagonalizesAVerify that PTAPgives the correct diagonal form. A=[4200240000420024]52E53E54E55E56E57E58E59EFind ATA and AAT for the matrix below. What do you observe? A=[132461]Finding Age Distribution Vectors In Exercises 1-6, use the age transition matrix L and age distribution vector x1 to find the age distribution vectors x2 and x3. Then find a stable age distribution vector. L=[02120],x1=[1010]2E3EFinding Age Distribution Vectors In Exercises 1-6, use the age transition matrix L and the age distribution vector x1 to find the age distribution vectors x2 and x3. Then find a stable age distribution vector. L=[02012000120],x1=[888]5E6EPopulation Growth Model A population has the characteristics below. a A total of 75 of the population survives the first year. Of that 75, 25 survives the second year. The maximum life span is 3 years. b The average number of offspring for each member of the population is 2 the first year, 4 the second year, and 2 the third year. The population now consists of 160 members in each of the three age classes. How many members will there be in each age class in 1 year? in 2 years?Population Growth Model A population has the characteristics below. a A total of 80 of the population survives the first year. Of that 80, 25 survive the second year. The maximum life span is 3 years. b The average number of offspring for each member of the population is 3 the first year, 6 the second year, and 3 the third year. The population now consists of 120 members in each of the three age classes. How many members will there be in each age class in 1 year? in 2 years?9EFind the limit if it exists of Anx1 as n approaches infinity, where A=[02120], and x1=[aa]11E12E13E14E15E16E17E18E19E20ESolving a System of Linear Differential Equations In Exercises 21-28, solve the system of first-order linear differential equations. y1=y14y2y2=2y2Solving a System of Linear Differential Equations In Exercises 21-28, solve the system of first-order linear differential equations. y1=y14y2y2=2y1+8y223ESolving a System of Linear Differential Equations In Exercises 21-28, solve the system of first-order linear differential equations. y1=y1y2y2=2y1+4y225E26ESolving a System of Linear Differential Equations In Exercises 21-28, solve the system of first-order linear differential equations. y1=3y25y3y2=4y14y2+10y3y3=4y328E29E30E31E32E33E34E35E36E37E38E39E40E41E42E43E44E45E46ERotation of a Conic In Exercises 45-52, use the Principle Axes Theorem to perform a rotation of axes to eliminate the xy-term the quadratic equation. Identify the resulting rotated conic and give its equation in the new coordinate system. 2x24xy+5y236=048E49E50E51E52E53E54E