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All Textbook Solutions for Elementary Linear Algebra (MindTap Course List)
74EProof Use the concept of a fixed point of a linear transformation T:VV. A vector u is a fixed point when T(u)=u. (a) Prove that 0 is a fixed point of a liner transformation T:VV. (b) Prove that the set of fixed points of a linear transformation T:VV is a subspace of V. (c) Determine all fixed points of the linear transformation T:R2R2 represented by T(x,y)=(x,2y). (d) Determine all fixed points of the linear transformation T:R2R2 represented by T(x,y)=(y,x).A translation in R2 is a function of the form T(x,y)=(xh,yk), where at least one of the constants h and k is nonzero. (a) Show that a translation in R2 is not a linear transformation. (b) For the translation T(x,y)=(x2,y+1), determine the images of (0,0,),(2,1), and (5,4). (c) Show that a translation in R2 has no fixed points.Proof Prove that a the zero transformation and b the identity transformation are linear transformations.Let S={v1,v2,v3} be a set of linearly independent vectors in R3. Find a linear transformation T from R3 into R3 such that the set {T(v1),T(v2),T(v3)} is linearly dependent.79EProof Let V be an inner product space. For a fixed vector v0 in V, define T:VR by T(v)=v,v0. Prove that T is a linear transformation.81E82E83E84EFinding the Kernel of a Linear Transformation In Exercises 1-10, find the kernel of the linear transformation. T:R3R3, T(x,y,z)=(0,0,0)Finding the Kernel of a Linear Transformation In Exercises 1-10, find the kernel of the linear transformation. T:R3R3, T(x,y,z)=(x,0,z)Finding the Kernel of a Linear Transformation In Exercises 1-10, find the kernel of the linear transformation. T:R4R4, T(x,y,z,w)=(y,x,w,z)Finding the Kernel of a Linear Transformation In Exercises 1-10, find the kernel of the linear transformation. T:R3R3, T(x,y,z)=(z,y,x)Finding the Kernel of a Linear Transformation In Exercises 1-10, find the kernel of the linear transformation. T:P3R, T(a0+a1x+a2x2+a3x3)=a1+a2Finding the Kernel of a Linear Transformation In Exercises 1-10, find the kernel of the linear transformation. T:P2R, T(a0+a1x+a2x2)=a0Finding the Kernel of a Linear Transformation In Exercises 1-10, find the kernel of the linear transformation. T:P2P1, T(a0+a1x+a2x2)=a1+2a2xFinding the Kernel of a Linear Transformation In Exercise 1-10, find the kernel of the linear transformation. T:P3P2T(a0+a1x+a2x2+a3x3)=a1x+2a2x2+3a3x3Finding the Kernel of a Linear Transformation In Exercises 1-10, find the kernel of the linear transformation. T:R2R2,T(x,y)=(x+2y,yx)Finding the Kernel of a Linear Transformation In Exercises 1-10, find the kernel of the linear transformation. T:R2R2,T(x,y)=(xy,yx)Finding the Kernel and Range In Exercises 11-18, define the linear transformation T by T(x)=Ax. Find a the kernel of T and b the range of T. A=[1234]Finding the Kernel and Range In Exercises 11-18, define the linear transformation T by T(x)=Ax. Find a the kernel of T and b the range of T. A=[1236]Finding the Kernel and Range In Exercises 11-18, define the linear transformation T by T(x)=Ax. Find a the kernel of T and b the range of T. A=[112012]Finding the Kernel and Range In Exercises 11-18, define the linear transformation T by T(x)=Ax. Find a the kernel of T and b the range of T. A=[121021]Finding the Kernel and Range In Exercises 11-18, define the linear transformation T by T(x)=Ax. Find a the kernel of T and b the range of T. A=[131322]Finding the Kernel and Range In Exercises 11-18, define the linear transformation T by T(x)=Ax. Find a the kernel of T and b the range of T. A=[111201]Finding the Kernel and Range In Exercises 11-18, define the linear transformation T by T(x)=Ax. Find a the kernel of T and b the range of T. A=[1214312143131211]Finding the Kernel and Range In Exercises 11-18, define the linear transformation T by T(x)=Ax. Find a the kernel of T and b the range of T. A=[132142350021210]Finding the Kernel, Nullity, Range, and Rank In Exercises 19-32, define the linear transformation Tby T(x)=Ax. Find a ker(T),bnullity(T),crange(T), and d rank(T). A=[1111]Finding the Kernel, Nullity, Range, and Rank In Exercises 19-32, define the linear transformation Tby T(x)=Ax. Find a ker(T),bnullity(T),crange(T), and d rank(T). A=[3296]Finding the Kernel, Nullity, Range and Rank In Exercises 19-32, define the linear transformation T by T(x)=Ax. Find a ker(T), b nullity(T), c range(T) and d rank(T). A=[531111]Finding the Kernel, Nullity, Range and Rank In Exercises 19-32, define the linear transformation T by T(x)=Ax. Find a ker(T), b nullity(T), c range(T) and d rank(T). A=[410023]Finding the Kernel, Nullity, Range and Rank In Exercises 19-32, define the linear transformation T by T(x)=Ax. Find a ker(T), b nullity(T), c range(T) and d rank(T). A=[910310310110]Finding the Kernel, Nullity, Range and Rank In Exercises 19-32, define the linear transformation T by T(x)=Ax. Find a ker(T), b nullity(T), c range(T) and d rank(T). A=[1265265262526]Finding the Kernel, Nullity, Range and Rank In Exercises 19-32, define the linear transformation T by T(x)=Ax. Find a ker(T), b nullity(T), c range(T) and d rank(T). A=[101010101]Finding the Kernel, Nullity, Range and Rank In Exercises 19-32, define the linear transformation T by T(x)=Ax. Find a ker(T), b nullity(T), c range(T) and d rank(T). A=[100000001]Finding the Kernel, Nullity, Range, and Rank In Exercises 19-32, define the linear transformation T by T(x)=Ax. Find a ker(T), b nullity(T), c range(T)and d rank(T). A=[494929494929292919]Finding the Kernel, Nullity, Range, and Rank In Exercises 19-32, define the linear transformation T by T(x)=Ax. Find a ker(T), b nullity(T), c range(T)and d rank(T). A=[132313231323132313]Finding the Kernel, Nullity, Range, and Rank In Exercises 19-32, define the linear transformation T by T(x)=Ax. Find a ker(T), b nullity(T), c range(T)and d rank(T). A=[0234011]Finding the Kernel, Nullity, Range, and RankIn Exercises 19-32, define the linear transformation T by T(x)=Ax. Find a ker(T), b nullity(T), c range(T), and d rank(T). A=[11000011]Finding the Kernel, Nullity, Range, and Rank In Exercises 19-32, define the linear transformation Tby T(x)=Ax. Find a ker(T), b nullity(T), c range(T), and d rank(T). A=[22311311111335014662416]32EFinding the Nullity and Describing the Kernel and Range In Exercises 33-40, let T:R3R3 be a linear transformation. Find the nullity of T and give a geometric description of the kernel and range of T. rank(T)=234E35EFinding the Nullity and Describing the Kernel and Range In Exercises 33-40, let T:R3R3 be a linear transformation. Find the nullity of T and give a geometric description of the kernel and range of T. rank(T)=337E38EFinding the Nullity and Describing the Kernel and Range In Exercises 33-40, let T:R3R3be a linear transformation. Find the nullity of T and give a geometric description of the kernel and range of T. T is the projection onto the vector v=(1,2,2): T(x,y,z)=x+2y+2z9(1,2,2)40EFinding the Nullity of a Linear Transformation In Exercises 41-46, find the nullity of T. T:R4R2, rank(T)=242EFinding the Nullity of a Linear TransformationIn Exercises 41-46, find the nullity of T. T:P5P2, rank(T)=3Finding the Nullity of a Linear TransformationIn Exercises 41-46, find the nullity of T. T:P3P1, rank(T)=2Finding the Nullity of a Linear TransformationIn Exercises 41-46, find the nullity of T. T:M2,4M4,2, rank(T)=446EVerifying That T Is One-to-One and Onto In Exercises 47-50, verify that the matrix defines a linear function T that is one-to-one and onto. A=[2002]Verifying That T Is One-to-One and Onto In Exercises 47-50, verify that the matrix defines a linear function T that is one-to-one and onto. A=[1001]Verifying That T Is One-to-One and Onto In Exercises 47-50, verify that the matrix defines a linear function T that is one-to-one and onto. A=[100001010]50E51E52E53EDetermining Whether T Is One-to-One, Onto, or Neither In Exercises 51-54, determine whether the linear transformation is one-to-one, onto, or neither. T:R5R3, T(x)=Ax, where A is given in Exercise 18 18. A=[132142350021210]Identify the zero element and standard basis for each of the isomorphic vector spaces in Example 12. EXAMPLE 12 Isomorphic Vector spaces The vector spaces below are isomorphic to each other. a. R4=4space b. M4,1=spaceofall41matrices c. M2,2=spaceofall22matrices d. P3=spaceofallpolynomialsofdegree3orless e. V={(x1,x2,x3,x4,0):xiisarealnumber} subspace of R5Which vector spaces are isomorphic to R6? a M2,3 b P6 c C[0,6] d M6,1 e P5 f C[3,3] g {(x1,x2,x3,0,x5,x6,x7):xiisarealnumber}Calculus Define T:P4P3 by T(p)=p. What is the kernel of T?Calculus Define T:P2R by T(p)=01p(x)dx What is the kernel of T?Let T:R3R3 be the linear transformation that projects u onto v=(2,1,1). (a) Find the rank and nullity of T. (b) Find a basis for the kernel of T.CAPSTONE Let T:R4R3 be the linear transformation represented by T(x)=Ax, where A=[121001230001]. (a) Find the dimension of the domain. (b) Find the dimension of the range. (c) Find the dimension of the kernel. (d) Is T one-to-one? Explain. (e) Is T is onto? Explain. (f) Is T an isomorphism? Explain.61E62E63E64E65E66EGuided Proof Let B be an invertible nn matrix. Prove that the linear transformation T:Mn,nMn,n represented by T(A)=AB is an isomorphism. Getting started: To show that the linear transformation is an isomorphism, you need to show that T is both onto and one-to-one. (i) T is a linear transformation with vector spaces of equal dimension, so by Theorem 6.8, you only need to show that T is one-to-one. (ii) To show that T is one-to-one, you need to determine the kernel of T and show that it is {0} Theorem 6.6. Use the fact that B is an invertible nn matrix and that T(A)=AB. (iii) Conclude that T is an isomorphism.68E69E70EThe Standard Matrix for a Linear TransformationIn Exercises 1-6, find the standard matrix for the linear transformation T. T(x,y)=(x+2y,x2y)The Standard Matrix for a Linear TransformationIn Exercises 1-6, find the standard matrix for the linear transformation T. T(x,y)=(2x3y,xy,y4x)The Standard Matrix for a Linear TransformationIn Exercises 1-6, find the standard matrix for the linear transformation T. T(x,y,z)=(x+y,xy,zx)The Standard Matrix for a Linear TransformationIn Exercises 1-6, find the standard matrix for the linear transformation T. T(x,y)=(5x+y,0,4x5y)The Standard Matrix for a Linear TransformationIn Exercises 1-6, find the standard matrix for the linear transformation T. T(x,y,z)=(3x2z,2yz)The Standard Matrix for a Linear Transformation In Exercises 1-6, find the standard matrix for the linear transformation T. T(x1,x2,x3,x4)=(0,0,0,0)Finding the Image of a Vector In Exercises 7-10, use the standard matrix for the linear transformation T to find the image of the vector v. T(x,y,z)=(2x+y,3yz), v=(0,1,1)Finding the Image of a Vector In Exercises 7-10, use the standard matrix for the linear transformation T to find the image of the vector v. T(x,y)=(x+y,xy,2x,2y), v=(3,3)Finding the Image of a Vector In Exercises 7-10, use the standard matrix for the linear transformation T to find the image of the vector v. T(x,y)=(x3y,2x+y,y), v=(2,4)Finding the Image of a Vector In Exercises 7-10, use the standard matrix for the linear transformation T to find the image of the vector v. T(x1,x2,x3,x4)=(x1x3,x2x4,x3x1,x2+x4), v=(1,2,3,2)Finding the Standard Matrix and the ImageIn Exercises 11-22, a find the standard matrix A for the linear transformation T, b use A to find the image of vector v, and c sketch the graph of v and its image. T is the reflection in the origin in R2: T(x,y)=(x,y), v=(3,4).Finding the Standard Matrix and the Image In Exercises 11-22, a find the standard matrix A for the linear transformation T, b use A to find the image of the vector v, and c sketch the graph of v and its image. T is the reflection in the line y=x in R2: T(x,y)=(y,x), v=(3,4).Finding the Standard Matrix and the Image In Exercises 11-22, a find the standard matrix A for the linear transformation T, b use A to find the image of the vector v, and c sketch the graph of v and its image. T is the reflection in the y-axis in R2: T(x,y)=(x,y), v=(2,3).14EFinding the Standard Matrix and the Image In Exercises 11-22, a find the standard matrix A for the linear transformation T, b use A to find the image of the vector v, and c sketch the graph of v and its image. T is the counterclockwise rotation of 45 in R2, v=(2,2).Finding the Standard Matrix and the ImageIn Exercises 11-22, a find the standard matrix A for the linear transformation T, b use A to find the image of vector v, and c sketch the graph of v and its image. T is the counterclockwise rotation of 120 in R2, v=(2,2).17E18E19E20EFinding the Standard Matrix and the Image In Exercise 11-22, a find the standard matrix A for the linear transformations T, b use A to find the image of the vector v, and c sketch the graph of v and its image. T is the projection onto the vector w=(3,1) in R2:T(v)=2projwv, v=(1,4).Finding the Standard Matrix and the Image In Exercise 11-22, a find the standard matrix A for the linear transformations T, b use A to find the image of the vector v, and c sketch the graph of v and its image. T is the reflection in the vector w=(3,1) in R2:T(v)=2projwvv, v=(1,4).Finding the Standard Matrix and the Image In Exercises 23-26, a find the standard matrix A for the linear transformation T and b use A to find the image of the vector v. Use a software program or a graphing utility to verify your result. T(x,y,z)=(2x+3yz,3x2z,2xy+z), v=(1,2,1)24E25E26EFinding Standard Matrices for CompositionsIn Exercises 27-30, find the standard matrices Aand Afor T=T2T1and T=T1T2. T1:R2R2, T1(x,y)=(x2y,2x+3y) T2:R2R2, T2(x,y)=(y,0)28EFinding Standard Matrices for Compositions In Exercises 2730, find the standard matrices A and A for T=T2T1 and T=T1T2. T1:R2R3,T1(x,y)=(2x+3y,x+y,x2y)T2:R3R2,T2(x,y,z)=(x2y,z+2x)Finding Standard Matrices for Compositions In Exercises 2730, find the standard matrices A and A for T=T2T1 and T=T1T2. T1:R2R3,T1(x,y)=(x,y,y)T2:R3R2,T2(x,y,z)=(y,z)Finding the Inverse of a Linear TransformationIn Exercises 31-36, determine whether the linear transformation in invertible. If it is, find its inverse. T(x,y)=(4x,4y)Finding the Inverse of a Linear TransformationIn Exercises 31-36, determine whether the linear transformation in invertible. If it is, find its inverse. T(x,y)=(2x,0)Finding the Inverse of a Linear TransformationIn Exercises 31-36, determine whether the linear transformation in invertible. If it is, find its inverse. T(x,y)=(x+y,3x+3y)34EFinding the Inverse of a linear TransformationIn Exercises 31-36, determine whether the linear transformation in invertible. If it is, find its inverse. T(x1,x2,x3)=(x1,x+x2,x1+x2+x3)Finding the Inverse of a Linear Transformation In Exercises 31-36, determine whether the linear transformation is invertible. If it is, find its inverse. T(x1,x2,x3,x4)=(x12x2,x2,x3+x4,x3)Finding the Image Two Ways In Exercises 37-42, find T(v)by using (a)the standard matrix and (b)the matrix relative to Band B. T:R2R3, T(x,y)=(x+y,x,y), v=(5,4), B={(1,1),(0,1)}, B={(1,1,0),(0,1,1),(1,0,1)}Finding the Image Two Ways In Exercises 37-42, find T(v)by using (a)the standard matrix and (b)the matrix relative to Band B. T:R3R2, T(x,y,z)=(xy,yz), v=(2,4,6), B={(1,1,1),(1,1,0),(0,1,1)}, B={(1,1),(2,1,)}Finding the Image Two Ways In Exercises 37-42, find T(v)by using (a)the standard matrix and (b)the matrix relative to Band B. T:R3R4, T(x,y,z)=(2x,x+y,y+z,x+z), v=(1,5,2), B={(2,0,1),(0,2,1),(1,2,1)}, B={(1,0,0,1),(0,1,0,1),(1,0,1,0),(1,1,0,0)}40E41EFinding the Image Two Ways In Exercises 37-42, find T(v)by using (a)the standard matrix and (b)the matrix relative to Band B. T:R2R2, T(x,y)=(3x13y,x4y), v=(4,8), B=B={(2,1),(5,1)}Let T:P2P3 be the linear transformation T(p)=xp. Find the matrix for T relative to the bases B={1,x,x2} and B={1,x,x2,x3}.Let T:P2P4 be the linear transformation T(p)=x2p. Find the matrix for T relative to the bases B={1,x,x2} and B={1,x,x2,x3,x4}.Calculus Let B={1,x,ex,xex} be a basis for a subspace W of the space of continuous functions, and let Dx be the differential operator on W. Find the matrix for Dx relative to the basis B.Calculus Repeat Exercise 45 for B={e2x,xe2x,x2e2x}. 45. Calculus Let B={1,x,ex,xex} be a basis for a subspace W of the space of continuous functions, and let Dx be the differential operator on W. Find the matrix for Dx relative to the basis B.Calculus Use the matrix from Exercise 45 to evaluate Dx[4x3xex]. 45. Calculus Let B={1,x,ex,xex} be a basis for a subspace W of the space of continuous functions, and let Dx be the differential operator on W. Find the matrix for Dx relative to the basis B.48ECalculus Let B={1,x,x2,x3} be a basis for P3, and T:P3P4 be the linear transformation represented by T(xk)=0xtkdt. (a) Find the matrix A for T with respect to B and the standard basis for P4. (b) Use A to integrate p(x)=84x+3x3.50EDefine T:M2,3M3,2 by T(A)=AT. aFind the matrix for T relative to the standard bases for M2,3 and M3,2. bShow that T is an isomorphism. cFind the matrix for the inverse of T.Let T be a linear transformation T such that T(v)=kv for v in Rn. Find the standard matrix for T.True or False? In Exercises 53 and 54, 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 T:RnRm is a linear transformation such that T(e1)=[a11,a21am1]TT(e2)=[a12,a22am2]TT(en)=[a1n,a2namn]T then the mn matrix A=[aij] whose columns corresponds to T(ei) is such that T(v)=Av for every v in Rn is called the standard matrix for T. b All linear transformations T have a unique inverse T1.54E55E56E57EWriting Look back at theorem 4.19 and rephrase it in terms of what learned in this chapter.Finding a Matrix for a Linear Transformation In Exercises 1-12, find the matrix Afor T relative to the basis B. T:R2R2:T(x,y)=(2xy,yx) B={(1,2),(0,3)}Finding a Matrix for a Linear Transformation In Exercises 1-12, find the matrix Afor T relative to the basis B. T:R2R2:T(x,y)=(2x+y,x2y) B={(1,2),(0,4)}3EFinding a Matrix for a Linear Transformation In Exercises 1-12, find the matrix Afor T relative to the basis B. T:R2R2:T(x,y)=(x2y,4x) B={(2,1),(1,1)}5E6E7EFinding a Matrix for a Linear Transformation In Exercises 1-12, find the matrix A for T relative to the basis B. T:R3R3,T(x,y,z)=(0,0,0), B={(1,1,0),(1,0,1),(0,1,1)}9EFinding a Matrix for a Linear Transformation In Exercises 1-12, find the matrix A for T relative to the basis B. T:R3R3,T(x,y,z)=(x,xy,yz), B={(0,1,2),(2,0,3),(1,3,0)}11E12E13ERepeat Exercise 13 for B={(1,1),(2,3)}, B={(1,1),(0,1)}, and [v]B=[13]T. Use matrix A in Exercise 13. Let B={(1,3),(2,2)} and B={(12,0),(4,4)} be bases for R2, and let A=[3204] be the matrix for T:R2R2 relative to B. a Find the transition matrix P from B to B. b Use the matrices P and A to find [v]B and [T(v)]B, where [v]B=[12]T. c Find P1 and A the matrix for T relative to B. d Find [T(v)]B two ways.15E16E17ERepeat Exercise 17 for B={(1,1,1),(1,1,1),(1,1,1)}B={(1,0,0),(0,1,0),(0,0,1)},and [v]B=[211]T. Use matrix A in Exercise 17. Let B={(1,1,0),(1,0,1),(0,1,1)} and B={(1,0,0),(0,1,0),(0,0,1)} be bases for R3, and let A=[321121221212152] be the matrix for T:R3R3 relative to B. a Find the transition matrix P from B to B. b Use the matrices P and A to find [v]B and [T(v)]B, where [v]B=[101]T. c Find P1 and A the matrix for T relative to B. d Find [T(v)]B two ways.Similar Matrices In Exercises 19-22, use the matrix P to show that the matrices A and Aare similar. P=[1112],A=[1272011],A=[1240]Similar Matrices In Exercises 19-22, use the matrix P to show that the matrices A and Aare similar. P=A=A=[11201]Similar Matrices In Exercises 19-22, use the matrix Pto show that the matrices Aand Aare similar. P=[500040003], A=[5100840096], A=[58010400126]Similar Matrices In Exercises 19-22, use the matrix Pto show that the matrices Aand Aare similar. P=[111011001], A=[500030001], A=[522032001]Diagonal Matrix for a Linear Transformation In Exercises 23 and 24, let Abe the matrix for T:R3R3relative to the standard basis. Find diagonal matrix Afor Trelative to the basis B. A=[020110001] B={(1,1,0),(2,1,0),(0,0,1)}Diagonal Matrix for a Linear Transformation In Exercises 23 and 24, let Abe the matrix for T:R3R3relative to the standard basis. Find diagonal matrix Afor Trelative to the basis B. A=[321121221212152] B={(1,1,1),(1,1,1),(1,1,1)}Proof Prove that if A and B are similar matrices, then |A|=|B|. Is the converse true?Illustrate the result of exercise 25 using the matrice A=[100020003], B=[1171010810181217] P=[110212111], P1=[112012123] where B=P1AP.27E28E29E30E31E32E33E34E35EProof Prove that if A and B are similar matrices and A is nonsingular, then B is also nonsingular and A1 and B1 are similar matrices.37E38E39E40E41E42E1E2E3E4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20EFinding Fixed Points of a Linear Transformation In Exercises 15-22, find all fixed points of the linear transformation. Recall that the vector vis a fixed point of Twhen T(v)=v. A horizontal shear.Finding Fixed Points of a Linear Transformation In Exercises 15-22, find all fixed points of the linear transformation. Recall that the vector vis a fixed point of Twhen T(v)=v. A vertical shear.23E24E25E26E27E28E29E30E31E32E33E34E35E36ESketching an Image of a Rectangle In Exercises 31-38, sketch the image of the rectangle with vertices at (0,0), (1,0), (1,2)and (0,2)under the specified transformation. T is the shear represented by T(x,y)=(x+y,y).Sketching an Image of a Rectangle In Exercises 31-38, sketch the image of the rectangle with vertices at (0,0), (1,0), (1,2)and (0,2)under the specified transformation. T is the shear represented by T(x,y)=(x,y+2x).39E40E41E42E43E44EGiving a Geometric Description In Exercises 45-50, give a geometric description of the linear transformation define by the elementary matrix. A=[2001]46E47E48E49EGiving a Geometric Description In Exercises 45-50, give a geometric description of the linear transformation defined by the elementary matrix. A=[14001]51E52E53E54E55E56E57E58E59E60E61E62E63E64E65E66E67E68E69EDetermining a matrix to produce a pair of rotation In Exercise 69-72, determine the matrix that produces the pair of rotations. Then find the image of the vector (1,1,1)under these rotation. 30 about z-axis and then 60 about y-axis71E72E1CRFinding an Image and a PreimageIn Exercises 1-6, find a the image of v and b the preimage of w for the linear transformation. T:R2R2, T(v1,v2)=(v1+v2,2v2), v=(4,1), w=(8,4).Finding an Image and a PreimageIn Exercises 1-6, find a the image of v and b the preimage of w for the linear transformation. T:R3R3, T(v1,v2,v3)=(0,v1+v2,v2+v3), v=(3,2,5), w=(0,2,5).4CRFinding an Image and a PreimageIn Exercises 1-6, find a the image of v and b the preimage of w for the linear transformation. T:R2R3, T(v1,v2)=(v1+v2,v1v2,2v1+3v2), v=(2,3), w=(1,3,4).6CRLinear Transformations and Standard Matrices In Exercises 7-18, determine whether the function is a linear transformation. If it is, find its standard matrix A. T:RR2, T(x)=(x,x+2).8CRLinear Transformations and Standard MatricesIn Exercises 7-18, determine whether the function is a linear transformation. If it is, find its standard matrix A. T:R2R2, T(x1,x2)=(x1+2x2,x1x2).Linear Transformations and Standard MatricesIn Exercises 7-18, determine whether the function is a linear transformation. If it is, find its standard matrix A. T:R2R2, T(x1,x2)=(x1+3,x2).Linear Transformations and Standard MatricesIn Exercises 7-18, determine whether the function is a linear transformation. If it is, find its standard matrix A. T:R2R2, T(x,y)=(x2y,2yx)12CRLinear Transformations and Standard MatricesIn Exercises 7-18, determine whether the function is a linear transformation. If it is, find its standard matrix A. T:R2R2, T(x,y)=(x+h,y+k), h0 or k0 translation in R2Linear Transformations and Standard MatricesIn Exercises 7-18, determine whether the function is a linear transformation. If it is, find its standard matrix A. T:R2R2, T(x,y)=(|x|,|y|)Linear Transformations and Standard MatricesIn Exercises 7-18, determine whether the function is a linear transformation. If it is, find its standard matrix A. 15.T:R3R3, T(x1,x2,x3)=(x1+x2,2,x3x1)16CRLinear Transformations and Standard MatricesIn Exercises 7-18, determine whether the function is a linear transformation. If it is, find its standard matrix A. T:R3R3, T(x,y,z)=(z,y,x)18CRLet T be a linear transformation from R2 into R2 such that T(2,0)=(1,1) and T(0,3)=(3,3). Find T(1,1) and T(0,1)Let T be a linear transformation from R3 into R such that T(1,1,1)=1, T(1,1,0)=2 and T(1,0,0)=3. Find T(0,1,1)Let T be a linear transformation from R2 into R2 such that T(4,2)=(2,2) and T(3,3)=(3,3). Find T(7,2).Let T be a linear transformation from R2 into R2 such that T(1,1)=(2,3) and T(0,2)=(0,8). Find T(2,4).Linear Transformation Given by a Matrix In Exercises 23-28, define the linear transformation T:RnRm by T(v)=Av. Use the matrix A to a determine the dimensions of Rn and Rm, b find the image of v, and c find the preimage of w. A=[012200], v=(6,1,1), w=(3,5)Linear Transformation Given by a Matrix In Exercises 23-28, define the linear transformation T:RnRm by T(v)=Av. Use the matrix A to a determine the dimensions of Rn and Rm, b find the image of v, and c find the preimage of w. A=[121101], v=(5,2,2), w=(4,2)Linear Transformation Given by a Matrix In Exercises 23-28, define the linear transformation T:RnRm by T(v)=Av. Use the matrix A to a determine the dimensions of Rn and Rm, b find the image of v, and c find the preimage of w. A=[111011001], v=(2,1,5), w=(6,4,2)Linear Transformation Given by a Matrix In Exercises 23-28, define the linear transformation T:RnRm by T(v)=Av. Use the matrix A to a determine the dimensions of Rn and Rm, b find the image of v, and c find the preimage of w. A=[2101], v=(8,4), w=(5,2).Linear Transformation Given by a Matrix In Exercises 23-28, define the linear transformation T:RnRm by T(v)=Av. Use the matrix A to a determine the dimensions of Rn and Rm, b find the image of v, and c find the preimage of w. A=[400511], v=(2,2), w=(4,5,0)Linear Transformation Given by a MatrixIn Exercises 23-28, define the linear transformation T:RnRmby T(v)=Av. Use the matrix A to a determine the dimensions of Rnand Rm, b find the image of v, and c find the preimage of w. A=[100113], v=(3,5), w=(5,2,1)Use the standard matrix for counterclockwise rotation in R2 to rotate the triangle with vertices (3,5), (5,3) and (3,0) counterclockwise 90 about the origin. Graph the triangles.Rotate the triangle in Exercise 29 counterclockwise 90 about the point (5,3). Graph the triangles. 29. Use the standard matrix for counterclockwise rotation in R2 to rotate the triangle with vertices (3,5), (5,3) and (3,0) counterclockwise 90 about the origin. Graph the triangles.Finding the Kernel and Range In Exercises 31-34, find a ker(T)and b range(T). T:R4R3, T(w,x,y,z)=(2w+4x+6y+5z,w2x+2y,8y+4z)Finding the Kernel and Range In Exercises 31-34, find a ker(T)and b range(T). T:R3R3, T(x,y,z)=(x+2y,y+2z,z+2x)Finding the Kernel and Range In Exercises 31-34, find a ker(T)and b range(T). T:R3R3, T(x,y,z)=(x,y,z+3y)Finding the Kernel and Range In Exercises 31-34, find a ker(T)and b range(T). T:R3R3, T(x,y,z)=(x+y,y+z,xz)Finding the Kernel, Nullity, Range, and Rank In Exercises 35-38, define the linear transformation T by T(v)=Av. Find a ker(T), b nullity(T), c range(T), and d rank(T). A=[121011]Finding the Kernel, Nullity, Range, and Rank In Exercises 35-38, define the linear transformation T by T(v)=Av. Find a ker(T), b nullity(T), c range(T), and d rank(T). A=[120122]Finding the Kernel, Nullity, Range, and Rank In Exercises 35-38, define the linear transformation T by T(v)=Av. Find a ker(T), b nullity(T), c range(T), and d rank(T). A=[213110013]Finding the Kernel, Nullity, Range, and Rank In Exercises 35-38, define the linear transformation T by T(v)=Av. Find a ker(T), b nullity(T), c range(T), and d rank(T). A=[111121010]For T:R5R3 and nullity(T)=2, find rank(T).For T:P5P3 and nullity(T)=4, find rank(T).For T:P4R5, and rank (T)=3, find nullity (T).42CR43CR44CR45CR46CRFinding Standard Matrices for Compositions In Exercises 47 and 48, find the standard matrices for T=T2T1and T=T1T2. T1:R2R3,T1(x,y)=(x,x+y,y)T2:R3R2,T2(x,y,z)=(0,y)