T is the linear o perator on R 5 de fined by [ Xtl [ -2r,

In l:.xercises 1-8. a linear operator Ton n3 and a basis 8 for n3 are gien. Compwe [Till- and determine whether B is a basis for n3 con.ri.rting of eigenllec/OrS ofT.

In l:.xercises 1-8. a linear operator Ton n3 and a basis 8 for n3 are gien. Compwe [Till- and determine whether B is a basis for n3 con.ri.rting of eigenllec/OrS ofT.

In l:.xercises 1-8. a linear operator Ton n3 and a basis 8 for n3 are gien. Compwe [Till- and determine whether B is a basis for n3 con.ri.rting of eigenllec/OrS ofT.

In l:.xercises 1-8. a linear operator Ton n3 and a basis 8 for n3 are gien. Compwe [Till- and determine whether B is a basis for n3 con.ri.rting of eigenllec/OrS ofT.

In l:.xercises 1-8. a linear operator Ton n3 and a basis 8 for n3 are gien. Compwe [Till- and determine whether B is a basis for n3 con.ri.rting of eigenllec/OrS ofT.

In l:.xercises 1-8. a linear operator Ton n3 and a basis 8 for n3 are gien. Compwe [Till- and determine whether B is a basis for n3 con.ri.rting of eigenllec/OrS ofT.

In l:.xercises 1-8. a linear operator Ton n3 and a basis 8 for n3 are gien. Compwe [Till- and determine whether B is a basis for n3 con.ri.rting of eigenllec/OrS ofT.

In l:.xercises 1-8. a linear operator Ton n3 and a basis 8 for n3 are gien. Compwe [Till- and determine whether B is a basis for n3 con.ri.rting of eigenllec/OrS ofT.

n Exercises 9- 20. a linear operator T on R " and its clwracterisric polynomial are given. Find, if possible, a basis for R" consisTing of eigemecrors ofT. If no such ~;: = ~; ]. t 2 + basis exists. explain

n Exercises 9- 20. a linear operator T on R " and its clwracterisric polynomial are given. Find, if possible, a basis for R" consisTing of eigemecrors ofT. If no such basis exists. explain 10 T

n Exercises 9- 20. a linear operator T on R " and its clwracterisric polynomial are given. Find, if possible, a basis for R" consisTing of eigemecrors ofT. If no such basis exists. explain 11 TR

n Exercises 9- 20. a linear operator T on R " and its clwracterisric polynomial are given. Find, if possible, a basis for R" consisTing of eigemecrors ofT. If no such basis exists. explain 12T

n Exercises 9- 20. a linear operator T on R " and its clwracterisric polynomial are given. Find, if possible, a basis for R" consisTing of eigemecrors ofT. If no such basis exists. explain T 13

n Exercises 9- 20. a linear operator T on R " and its clwracterisric polynomial are given. Find, if possible, a basis for R" consisTing of eigemecrors ofT. If no such basis exists. explain 14 T

n Exercises 9- 20. a linear operator T on R " and its clwracterisric polynomial are given. Find, if possible, a basis for R" consisTing of eigemecrors ofT. If no such basis exists. explain 15 T

n Exercises 9- 20. a linear operator T on R " and its clwracterisric polynomial are given. Find, if possible, a basis for R" consisTing of eigemecrors ofT. If no such basis exists. explain 16 T

n Exercises 9- 20. a linear operator T on R " and its clwracterisric polynomial are given. Find, if possible, a basis for R" consisTing of eigemecrors ofT. If no such basis exists. explain 18 T

n Exercises 9- 20. a linear oper ator T on R " and its clwracterisric polynomial are given. Find, if possible, a basis for R" consisTing of eigemecrors ofT. If no such basis exists. explain 18 T

n Exercises 9- 20. a linear operator T on R " and its clwracterisric polynomial are given. Find, if possible, a basis for R" consisTing of eigemecrors ofT. If no such basis exists. explain 1*9 T

n Exercises 9- 20. a linear operator T on R " and its clwracterisric polynomial are given. Find, if possible, a basis for R" consisTing of eigemecrors ofT. If no such basis exists. explain 20 T

In Exercises 21- 28, a linear operator Ton n is given. Find, if possible. a basis B for 'R" such thar 21 [T ]

In Exercises 21- 28, a linear operator Ton n is given. Find, if possible. a basis B for 'R" such thar 22 [T ]

In Exercises 21- 28, a linear operator Ton n is given. Find, if possible. a basis B for 'R" such thar 23 [T ]

In Exercises 21- 28, a linear operator Ton n is given. Find, if possible. a basis B for 'R" such thar 24 [T ]

In Exercises 21- 28, a linear operator Ton n is given. Find, if possible. a basis B for 'R" such thar 25 [T ]

In Exercises 21- 28, a linear operator Ton n is given. Find, if possible. a basis B for 'R" such thar 26 [T ]

In Exercises 21- 28, a linear operator Ton n is given. Find, if possible. a basis B for 'R" such thar 27 [T ]

In Exercises 21- 28, a linear operator Ton n is given. Find, if possible. a basis B for 'R" such thar 28 [T ]

f a linear operator on n" is diagonalizable. then its standard matrix is

For every linear operator on n. there is a bttsis B for n such that [T]G

A linear operator on n is diagonalizablc if and only if its standard ma

If T is a diagonalizable linear operator on n. there is a unique ba

lf T is a diagonalizable linear operator on n. there is a unique d

et W be a 2-dimcnsional subspace of n 3. The reflection of n

Let W be a 2-dimcnsional subspace of nJ The reflection of n 3 abou

If T is a linear operator on n" and B is a basis for n such that [T]6 is a diagonal matrix, then 8 consists of eigenvectors of

The characteristic polynomial of a linear operator T on 1?." is a poly

If the characteristic polynomial of a linear operator T on 1?." factors into a product of linear factors, then T is diagonalizable.

If the characteristic polynomial of a linear operator T on n" does not factor into a product of linear factors. then T is not di

If, for each eigenvalue A of a linear operator T on n, the dimension of the eigcnspacc ofT corresponding to A ec1uals the multiplicity of A, then T is diagonalizable. 41. Let IV be

Let IV be a two-dimensional subspace of n3 If Tw : n 3 - n 3 is the reflection of n 3 about W . then each nonzero vector in W is an eigenvector of Tw corresponding to the eigenvalue -

Let W be a two-dimensional subspace of n 3. If Tw: n 3 - n 3 is the reflection ofn3 about W.then each nonzero vector

If T is a diagonalizable linear operator having 0 as an eigenvalu

T is a linear operator on 1?.", then the sum of the multiplicitie

If T is a diagonalizable linear operator on n", then the sum of the multiplici

If T is a linear operator on n having 11 distinct eigenvalues. then T is

If 8 1 8 2, . , Bk are bases for distinct eigenspaces of a linear op

f 8,. 82 ..... Bt arc bases for all the distinct eigenspaces of a linear operator T, then 8 1 U Bz U U Bt is a ba~is for 1?." consist

n Exerci. 49-58 inet1r operaTor tmd irs characrerisric polynomial are given. DeTermine all The values of the scalar c for which the given linear operaTor OJI n3 is 110 49 T

n Exerci. 49-58 inet1r operaTor tmd irs characrerisric polynomial are given. DeTermine all The values of the scalar c for which the given linear operaTor OJI n3 is 110 50 T

n Exerci. 49-58 inet1r operaTor tmd irs characrerisric polynomial are given. DeTermine all The values of the scalar c for which the given linear operaTor OJI n3 is 110 51 T

n Exerci. 49-58 inet1r operaTor tmd irs characrerisric polynomial are given. DeTermine all The values of the scalar c for which the given linear operaTor OJI n3 is 110 52 T

n Exerci. 49-58 inet1r operaTor tmd irs characrerisric polynomial are given. DeTermine all The values of the scalar c for which the given linear operaTor OJI n3 is 110 53 T

n Exerci. 49-58 inet1r operaTor tmd irs characrerisric polynomial are given. DeTermine all The values of the scalar c for which the given linear operaTor OJI n3 is 110 54

n Exerci. 49-58 inet1r operaTor tmd irs characrerisric polynomial are given. DeTermine all The values of the scalar c for which the given linear operaTor OJI n3 is 110 55 T

n Exerci. 49-58 inet1r operaTor tmd irs characrerisric polynomial are given. DeTermine all The values of the scalar c for which the given linear operaTor OJI n3 is 110 56 TT

n Exerci. 49-58 inet1r operaTor tmd irs characrerisric polynomial are given. DeTermine all The values of the scalar c for which the given linear operaTor OJI n3 is 110 57 T

n Exerci. 49-58 inet1r operaTor tmd irs characrerisric polynomial are given. DeTermine all The values of the scalar c for which the given linear operaTor OJI n3 is 110 58

In Exercises 59- 64, the equation of a plane W through the origin of n 3 is given. Determine an explicit formula for the reflection Tw ofn3 about IV 59. x +y +z =

In Exercises 59- 64, the equation of a plane W through the origin of n 3 is given. Determine an explicit formula for the reflection Tw ofn3 about IV 60. 2x + y + z = 0

In Exercises 59- 64, the equation of a plane W through the origin of n 3 is given. Determine an explicit formula for the reflection Tw ofn3 about IV 6 1. X+ 2y- Z = 0 63. x + 8y - 5z = 0

In Exercises 59- 64, the equation of a plane W through the origin of n 3 is given. Determine an explicit formula for the reflection Tw ofn3 about IV

In Exercises 59- 64, the equation of a plane W through the origin of n 3 is given. Determine an explicit formula for the reflection Tw ofn3 about IV

In Exercises 59- 64, the equation of a plane W through the origin of n 3 is given. Determine an explicit formula for the reflection Tw ofn3 about IV 64. 3x - 4y + 5z = 0

In Exercises 65 and 66. the standard matrix of a reflection ofn3 about a 2-dimensional subspace IV is giwn. Find an equation for IV. 8 1 [ 1 9 -4 -8 I -4

In Exercises 65 and 66. the standard matrix of a reflection ofn3 about a 2-dimensional subspace IV is giwn. Find an equation for IV. - -2 3 -1 -2 -I -

Let IV be a 2-dimensional subspace of n3 (a) Prove that there exists a basis 8 for n3 such that [Uwle = [~ ~ n (b) Prove that [Tw ]e = G ~ basis in (a). ~]. - 1 (c) Prove that [Uw ]G = !

In Exercises 68-74, find an explicit fomwla for the orthogonal projection Uw of n 3 on the plane IV in each specified exerci.ve. Use Exercise 67(c) and the ba.vis obtained in the .vpec ijied exercise to obu1in your answer. Exercise 60

In Exercises 68-74, find an explicit fomwla for the orthogonal projection Uw of n 3 on the plane IV in each specified exerci.ve. Use Exercise 67(c) and the ba.vis obtained in the .vpec ijied exercise to obu1in your answer. Exercise 61

In Exercises 68-74, find an explicit fomwla for the orthogonal projection Uw of n 3 on the plane IV in each specified exerci.ve. Use Exercise 67(c) and the ba.vis obtained in the .vpec ijied exercise to obu1in your answer. Exercise 62

In Exercises 68-74, find an explicit fomwla for the orthogonal projection Uw of n 3 on the plane IV in each specified exerci.ve. Use Exercise 67(c) and the ba.vis obtained in the .vpec ijied exercise to obu1in your answer. Exercise 63

In Exercises 68-74, find an explicit fomwla for the orthogonal projection Uw of n 3 on the plane IV in each specified exerci.ve. Use Exercise 67(c) and the ba.vis obtained in the .vpec ijied exercise to obu1in your answer. Exercise 64

In Exercises 68-74, find an explicit fomwla for the orthogonal projection Uw of n 3 on the plane IV in each specified exerci.ve. Use Exercise 67(c) and the ba.vis obtained in the .vpec ijied exercise to obu1in your answer. Exercise 65

In Exercises 68-74, find an explicit fomwla for the orthogonal projection Uw of n 3 on the plane IV in each specified exerci.ve. Use Exercise 67(c) and the ba.vis obtained in the .vpec ijied exercise to obu1in your answer. Exercise 66

Let [u. v. wf be a basis for n3 and let T be the linear operator on n3 defined by T(au + bv + cw) = a u + b v for all scalars a, b, and c. (a) Find the eigenvalues of T and determine a basis for each eigenspace. (b) Is T diagonalizable? Justify your answer. 76. Let (u. v. wf be

Let (u. v. wf be a basis for n3 and let T be the linear operator on n3 defined by T(a u + b' ' + c w) = a u + b'' - cw for all scalars a, b, and c. (a) Find the eigenvalues of T and determine a basis for each eigenspace. (b) Is T d iagonaliz

Let T be a linear operator on n ru1d 8 be a basis for n" such that [T]e is a diagonal matrix. Prove that 8 must consist of eigenvectors of T

If T and U are diagonalizable linear operators on n", must T + U be a diagonalizable linear operator on n"? Justify your answe

If T is a diagonalizable linear operator on n". must cT be a diagonalizable linear operator on 'R." for any scalar c? Justify y

If T and U are diagonalizable linear operators o n 'R.", must TV be a diagonalizable linear operator on n"? Justify you

Let T be a linear operator on n". and suppose that v1 v2 ..... Vt arc eigenvectors of T con es pond ing to distinct

Let T and U be diagona lizable linear operators on n". Prove that if there is a basis for n" consisting o f eigenvectors of

Let T and U be linear operators on 'R.". If T 2 = U (where T 2 = 7T), then T is called a square root of U. S how that if U is diagonalizablc and has only nonnegative eigenvalues, the

et T be a li near operator on n and 6 1 62 ..... 8t be bases for a

T is the linear operator on R 5 defi ned by [ x1] [-I Lt1 - 9x2 + 13x3 + 18x4 - 9xsl x2 6.r1 + Sx2 - 6x3 - 8x4 + 4xs T x3 = 6x1 + 3x2 - 4x3 - 6x4 + 3xs . x4 - 2r

T is the linear o perator on R 5 de fined by [ Xtl [ -2r, - 4x2- 9x3 - 5x4 - 16x5 ] x2 x1 + 4x2 + 6x3 + 5x + 12rs T x3 = 4x1 + IOx2 + 20x3 + 14x4 + 37x5 x, 3x, + 2r2 + 3x3 + 2x + 4xs xs -