#3 please 4:241. Consider the matrices.mtaylor.web.unc.edu1 0 12A₁ 02- (₁ ¦ ¦) + - ( ¦ ¦) ^ - ( ¹ )0A₂ = 020, A33 1-1 0 -10-2ExercisesCompute the eigenvalues and eigenvectors of each Aj.2. Find the minimal polynomial of A; and find a basis of generalized eigen-vectors of Aj.3. Consider the transformation D: P2 P2 given by (1.4.15). Find theeigenvalues and eigenvectors of D. Find the minimal polynomial of D andfind a basis of P2 consisting of generalized eigenvectors of D.4. Suppose V is a finite dimensional complex vector space and T: V→V.Show that V has a basis of eigenvectors of T if and only if all the roots ofthe minimal polynomial mr(A) are simple.5. In the setting of (2.2.3)-(2.2.4), given SE L(V), show thatST=TS⇒S: GE(T, Aj) →GE(T, Aj).Exercises6. Show that if V is an n-dimensional complex vector space, S, T = L(V),and ST TS, then V has a basis consisting of vectors that are simultane-ously generalized eigenvectors of T and of S.Hint. Apply Proposition 2.2.6 to S: GE(T, X₁)→GE(T, A₁).7. Let V be a complex n-dimensional vector space, and take TE L(V), withminimal polynomial mȚ(A), as in (2.2.13). For l {1,..., K}, setmT(X)P(X) =x-deShow that, for each le {1,..., K}, there exists we V such that ve =Pe(T) we 0. Then show that (TAI)ve = 0, so one has a proof ofProposition 2.1.1 that does not use determinants.718. In the setting of Exercise 7, show that the exponent k; in (2.2.14) is thesmallest integer such that(TAI) annihilates GE (T, Aj).Hint. Review the proof of roposition 2.2.4. respect to the basis S, then(1.4.14)Ã-C-¹AC.REMARK. We say that n x n matrices A and A, related as in (1.4.14), aresimilar.EXAMPLE. Consider the linear transformationmtaylor.web.unc.edu(1.4.15)With respect to the basis(1.4.17)D: P₂ P₂, Df(x) = f'(x).(1.4.16)D has the matrix representation/0 1 0A 0020 0 0since Dey=0, Dv₂=₁, and Des=202. With respect to the basis-1, ₂-1+z₁ =1+z+2²,(1.4.19)1, 22, 2²,(1.4.18)D has the matrix representation26/0 1 -1A-0023)0 0 0.since De = 0, D₂, and Des= 1+ 2z = 2₂-0. The reader isinvited to verify (1.4.14) for this example.1. Vector spaces, linear transformations, and matricesExercises1. Consider T: P₂ → P2, given by Tp(x)=z¹fp(y) dy. Compute thematrix representation B of T with respect to the basis (1.4.16). ComputeAB and BA, with A given by (1.4.17).2. In the setting of Exercise 1, compute DT and TD on P₂ and comparetheir matrix representations, with respect to the basis (1.4.16), with ABand BA.3. In the setting of Exercise 1, take a ER and define(1.4.20)Tap(x)=P(y) dy, Ta: P₂P₂.Compute the matrix representation of Ta with respect to the basis (1.4.16).4. Compute the matrix representation of Ta, given by (1.4.20), with respectto the basis of P₂ given in (1.4.18).5. Let A: C² C² be given by4-(4)(with respect to the standard basis). Find a basis of C² with respect towhich the matrix representation of A is- (85).6. Let V (a cost+bsint: a,b C), and considerD=:V V.D-Compute the matrix representation of D with respect to the basis (cost, sint).7. In the setting of Exercise 6, compute the matrix representation of D withrespect to the basis (e",e-it). (See Exercise 4 of $1.3.)

Question

#3 please 4:241. Consider the matrices.mtaylor.web.unc.edu1 0 12A₁ 02- (₁ ¦ ¦) + - ( ¦ ¦) ^ - ( ¹ )0A₂ = 020, A33 1-1 0 -10-2ExercisesCompute the eigenvalues and eigenvectors of each Aj.2. Find the minimal polynomial of A; and find a basis of generalized eigen-vectors of Aj.3. Consider the transformation D: P2 P2 given by (1.4.15). Find theeigenvalues and eigenvectors of D. Find the minimal polynomial of D andfind a basis of P2 consisting of generalized eigenvectors of D.4. Suppose V is a finite dimensional complex vector space and T: V→V.Show that V has a basis of eigenvectors of T if and only if all the roots ofthe minimal polynomial mr(A) are simple.5. In the setting of (2.2.3)-(2.2.4), given SE L(V), show thatST=TS⇒S: GE(T, Aj) →GE(T, Aj).Exercises6. Show that if V is an n-dimensional complex vector space, S, T = L(V),and ST TS, then V has a basis consisting of vectors that are simultane-ously generalized eigenvectors of T and of S.Hint. Apply Proposition 2.2.6 to S: GE(T, X₁)→GE(T, A₁).7. Let V be a complex n-dimensional vector space, and take TE L(V), withminimal polynomial mȚ(A), as in (2.2.13). For l {1,..., K}, setmT(X)P(X) =x-deShow that, for each le {1,..., K}, there exists we V such that ve =Pe(T) we 0. Then show that (TAI)ve = 0, so one has a proof ofProposition 2.1.1 that does not use determinants.718. In the setting of Exercise 7, show that the exponent k; in (2.2.14) is thesmallest integer such that(TAI) annihilates GE (T, Aj).Hint. Review the proof of roposition 2.2.4. respect to the basis S, then(1.4.14)Ã-C-¹AC.REMARK. We say that n x n matrices A and A, related as in (1.4.14), aresimilar.EXAMPLE. Consider the linear transformationmtaylor.web.unc.edu(1.4.15)With respect to the basis(1.4.17)D: P₂ P₂, Df(x) = f'(x).(1.4.16)D has the matrix representation/0 1 0A 0020 0 0since Dey=0, Dv₂=₁, and Des=202. With respect to the basis-1, ₂-1+z₁ =1+z+2²,(1.4.19)1, 22, 2²,(1.4.18)D has the matrix representation26/0 1 -1A-0023)0 0 0.since De = 0, D₂, and Des= 1+ 2z = 2₂-0. The reader isinvited to verify (1.4.14) for this example.1. Vector spaces, linear transformations, and matricesExercises1. Consider T: P₂ → P2, given by Tp(x)=z¹fp(y) dy. Compute thematrix representation B of T with respect to the basis (1.4.16). ComputeAB and BA, with A given by (1.4.17).2. In the setting of Exercise 1, compute DT and TD on P₂ and comparetheir matrix representations, with respect to the basis (1.4.16), with ABand BA.3. In the setting of Exercise 1, take a ER and define(1.4.20)Tap(x)=P(y) dy, Ta: P₂P₂.Compute the matrix representation of Ta with respect to the basis (1.4.16).4. Compute the matrix representation of Ta, given by (1.4.20), with respectto the basis of P₂ given in (1.4.18).5. Let A: C² C² be given by4-(4)(with respect to the standard basis). Find a basis of C² with respect towhich the matrix representation of A is- (85).6. Let V (a cost+bsint: a,b C), and considerD=:V V.D-Compute the matrix representation of D with respect to the basis (cost, sint).7. In the setting of Exercise 6, compute the matrix representation of D withrespect to the basis (e&#34;,e-it). (See Exercise 4 of $1.3.)

Accepted Answer

The given transformation is D:P2→P2 defined by Df(x)=f'(x).
To Find: Eigenvalues, eigenvectors, and…

3. Consider the transformation D: P2 P2 given by (1.4.15). Find the eigenvalues and eigenvectors of D. Find the minimal polynomial of D and find a basis of P2 consisting of generalized eigenvectors of D.