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[answered] Math 115A: Linear Algebra Homework 7: Final version: Due Mo

i just need answer of 4 problem( 1. problem1 ?/ 2. problem2 ?/ 3. double star 15 / ?4. double star 16) . plz show me details of every steps.

Math 115A: Linear Algebra

Homework 7: Final version: Due Monday November 14

?5.2 2 b, c, d, f. 3d, e, f, 8, 12,

?6.1 3, 5, 11, 12, 17, 19, 20b, 23.

?6.2 2b, c, i, j, 3, 6, 7, 15 a, 16a.

If

P (x) = cn xn + cn?1 xn?1 + . . . + c0

is a polynomial and T : V ? V is a linear transformation, we define

P (T ) = cn T n + cn?1 T n?1 + . . . + c0 I,

where I : V ? V is the identity. (See Definition in Appendix E pg 563.

You can assume the theorems after the Definitions.) The Cayley-Hamilton

theorem states that if V is a finite dimensional and p(?) is the characteristic

polynomial of T , then

p(T ) = 0

Problem 1: Prove the Cayley-Hamilton theorem if dim V = 2. To do this,

pick a basis B = {?1 , ?2 }, let

A = [T ]B = a11 a12

a21 a22 ! and compute the characteristic polynomial p(?) of A and then calculate p(A).

It should all cancel out.

Problem 2: Show that if ? is an eigenvalue of T : V ? V , then ? + a is

an eigenvalue of T + aI, where I is the identity. More generally, if V is

finite dimensional and the characteristic polynomial of T is p(?), then the

characteristic polynomial of T ? aI is p(? + a)

Double star 14. Suppose dim V = 2 and that T : V ? V. Suppose that

the characteristic polynomial of T is (? ? ?1 )(? ? ?2 ). Show that either T is diagonalizable or we can find a basis B = {?1 , ?2 } so that

[T ]B = ?1 1

0 ?1 ! Hints: If ?1 6= ?2 , we can diagonalize. If ?1 = ?2 , let N = T ? ?1 I. Use

Cayley-Hamilton to show N 2 = 0 and apply double star 10 to N , which you

may assume. You can use problem 2 above.

Double star 15: Suppose V is a vector space with inner product h , i. If W

is a subspace of V , we define

W ? = {?|h?, ?i = 0 for all ? ? W }

If U is another subspace, show

(U + W )? = U ? ? W ?

Double star 16: Let V be an inner product space with orthonormal basis

B = {?1 , ?2 , . . . , ?n }.Let T : V ? V be linear. Let A = [T ]B . Show

Aji = hT (?i ), ?j i

Challenge problem 7.1 Suppose dim V = n and that T : V ? V is diagonalizable. Let p(?) is the characteristic polynomial of T . Show

p(T ) = 0

Challenge 7.2 Suppose dim V = 3 and that T : V ? V is non-zero. Suppose

the characteristic polynomial p(?) of T is

p(?) = ??3 .

Show that we can find a basis B = {?1 , ?2 , ?3 } so that [T ]B is either 0 0 0 1 0 0 0 0 0 or 0 0 0 1 0 0 0 1 0

Hint: See double star 13 and challenge 6.2. You may use Cayley-Hamilton.

Challenge 7.3 Let V be the vector space of all continuous real valued functions

on R. Define a linear transformation

T (f )(x) = Z x f (t) dt 0 So T (f ) is the antiderivative of f which vanishes at 0. Show T has no

eigenvectors. You can use the fundamental theorem of calculus and you will

need to remember some differential equations.

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