7/26/2019 115A Practice Final Problems

1/2

Math

115A: Sample

final

exam

Sections

1 and 3. Instructor: James

Fteitag

For

the

exam,

you

may

use

one

8

inch by 11 inch

(normal

sized

paper) piece

of

paper

with

anything

at all written

on

one side -

theorems,

exa,rnple

problems,

inspirational sayings

-

anything

goes.

There

will

be

8

problems

on

the

finaJ.

The

difficulty

will be on

the

level

of the

exa,rrs.

Keep in

mind this

sa,rrple

review

is

not

comprhensilre.

I will

post

more

problems

throughout

the week.

Problem

1 Eigenvalues

Ler0e(0,r/2).Ler

^

_

(

as(0)

-sin(d) \

^

-

\

stn(d) cos(o)

)'

-jlbtrffi*of

4 are-in*r-se

we

esn-rega*&A

as-i

complex

nrrmbsls,

Are there aay eigenvectors

over

IR? Explain why not intuitively.

Calculate the

eigenvalues and eigenvectors

over

C.

Problem

2

Some

basics

Let

/l

2 3\

A:12

3

4l.

\3

4

5/

Find

a

basis

of

N(,4). Find a basis

of

A(A). Diagonalize

-4.

Problem

3 A subspace

Prove that the set of

aJl

functions

that

can be

written in

the form a.sin(x

+

b)

for

a

C

and

b

e

IR

is

a

vector

sDace.

Is

it

finite

dimensiona,l?

Problem 4 Fbom class

Fliday

Let

S

:

U

-+

V,,

T :

V

-+

W

be

linea.r

maps

of finite

dirnensional

vector

spaces.

Suppose

that ?,9 is bijective. Prove that ,S is surjective

if

and

only ?

is

injective.

7/26/2019 115A Practice Final Problems

2/2

Problem

5 Use

elementary matrices?

Let A

e

Mn*n([')

be an

invertible matrix,

and

let

B

be any

other

matrix of.

Mnrn(F).

Prove

that

det(AB)

-

det(A)

.

det(B).

Pr6blem 6

A

map with

specified kernel

Let V

be

a

finite dimensional innfer

product

space. Let

W

be a subspace. Constuct

a

linear

operator ,S on

y

with lf(S)

-

Wr

and

R(,S)

:

W.

Problem

7

Using a map

with

specified kernel

Let

V be a finite dimensional innler

product

space.

Let

W

be

a subspace.

Use,

the

previous

problem

to

prove

that

di,m(W)

+

di,m(Wt)

-

di,m(V).

Problem

8

Examples or lack

thereof

Give an example of a 2

x

2 matrix

M over IR

such that

M has no

eigenvalues

in IR.

Can

you

give

an exarnple

of

a 3

x

3

matrix

M over

IR

such that

M

has no eigenvalues

in

lR?

Problem

I Representati"[ ir

f6[1"" than workin*

il" hand

Find

a

polynomial

q

e

P3(R)

such that

/1\ f'

,

\,

\,

,

\i)

:

Jo

P@)q(r)dr

for

allp

PB(R).

You

can

write

down

your polynomial

in

terms

of

an

inner

product.

Lnoan

Ar,cpeRa

Math 115A.