Name: ID Number:

No calculators or ang other deaices alloued,If ang quest'i,on i.s not clea,r, ask for clari,fi.cat'i,on.No cred.i,t ui,II be g'i.aen for i.llegi.ble solutions.If gou present d.i,fferent answers for the sa,rne problern,the uorst arlsurer ui,ll be graded.Show all your work. lBou your ansuters.l

MTH 4L5Exam LJuIy 21-, 201050 minutesSects: L.'J.-L.4, 2.L-2.6

1. 1tS points) Express matrix A as a sum of a symmetric matrix and a skew-symmetricmatrix, where

[ r2Jf4: l+ s 61.

lz8el

7S" d '+A -r"4-

A

r i r z 3l i lq Tl , iei ' l ; ? i j - [ l i ; i r " { L: ,b] I+l l .

l /'t i fptz 3 I I rq:*7, io *L ' t l I!q b d iFi ; $ g{J, i le o -LiL? g q j iB 6 ?l ' Lq p o i

s i (

-g

6 ra jro t t t IlLt , f J

[ , 3

l3 *la +

T: ,{---{s*

2. OS points) Determine which of the following functions ?, ,5 : 1R2 -- R2 is iinear:

r (1", \ : I r , ,1. sf [ , ' l ) : f , , -* ' , .1'\1" ' .1 )- lsrprt \Lz2l l Ls ' ' JIf a function is linear, give a proof; if a function is not iinear, show it with an example.

T/o-x\ >\*rD

c[ Xe

'tt},' &f,f

lx l3'4ib ry

T- rx,}

IItJ

I, l { }lL lI i It lt llFI

rdt l ;TJ," f :

lulr*obj X+ bx j j

rx+}bj iI

.=: o\ it,*41 r b i Yr * Ya ?LrFr I i $f f J

# rl*t l NuT

'{i!* e u"r*a

tI.t

I

|.I

I,I,I

\ / e#+ by) s S ( i l ; ; l?[

{*x,*b{ )+s f*r,

o* gx,*rr l *b

*nX* +bB{

'!\j i

ftbt

IL

{II

{t,t

L

I r ' ,r pgr0 / r .e\ , t \ / ^ . \ I= cL \ [x)+ b S(t) |

l], o"o matrix X solution(20 points) Given the matrices A :

of the matrix equation

A{x,4-d,k i e -? AB

YA-(>( * * f b

x {A- 6r\ . -8 D A-(,y, {*

13 -11 th 11| ' ^ l&r loD:1L-r 3.1 -- ' " - L3-6AX*BAB:0.

3.

AXA

bl

,- tj

: l

* t?

t lIa. l3l

drf fi")* tAI

-i-3

{a " , , r ) ' , 6 ! '1L' I I

, ' l r* ' i is r ' i;. .*- : t

j -6 I : t 3 jJ

l l t-.1. I- l

- {-,

*f

I -gf ,- tn{a*6t i 'X;

r.

i i aJ r+L, i -* ' ii - { t g) i t -s li3 1i l

- * f t

f " - r -b iI

i i' a ' t ii '?i . - l

r.' J= 1l E

Lb t I)

I

t

(20 points) Consider the matrices '

f r 2zfR: | 3 2 11,

L-3 2 7J

Verify whether the foliowing equations hold:

4.

FA

N(A) : .^/(B)?, l/(A') : l /(B')?, n(A) :,R(BX'

Justify your answers.

R(A') : E(B')?.

* r l2 l 'o l

E 13f E4

[r oolB:10 101.

L3 -2 0l

2-e l l r o, Q. i -o l o to o I Lo o

2_-Llg

o-. fi :[-s t

{\=

F

l la

lovo'lto i

ot}J

5 l- r l * 'l6 J

!IT5,

I,I

,!tIa6

il(Ar\ *

E"{ A) ,

IttIIt,

I

t*

Tr 3 -Rl1o-, s l " 'fo -r tb t

;%'o slr- l to ol

l ' Bit

Ll3 |

= f ,I :

Att

s l l rr [ * ' f . ,+ I Loo P II o l? DJ

iro'r>lulL0 0fr

l3

IlB-3 I i r o]J Gl" t - . l - l * ' i o r-a[ ' tA,L"o ol L"e 'o l

' r -1l{J2 lIT\J

/Yfe)

E(fs)4f,' F-"RT

5. (tS points) Determine whether the following statement true or false:

"Given a matrix A € IF-'", it holds that tr (A'A) :0 if and oniy if A:0."

If the statement is true, prove it; if the statement is false, give an example showing it.

-D , rDhrhL{ A€lK

t,{l',a) = t{ *),',h

trt

/+r

i [4 'A] , , . , =i " ' II{t*-**S*

4o,o

/* , ' 4*,

Aur' A,aI.Y15'

.4Wzt I

i.t

n

Jk:r

ft'),o

c.7orJ

A), u

!- o loaJ

J".l"t- k'

l\IA=l

,h

)i ' r t

3/^t ' r I

'+

*

s3

+

L

t'

o

LrL

III

- tII1IL_

)

*o

d

?IlI

J

,7tJ

dr

-o

€

lD

o[

+

A

f-lc

It'-

hT;7iIt-l

: " f , .22 (a*, ' )VLzt '

t;) t!:' 3()

'+&+ b' * , r l '

/*)

l? Ae {

Ef^',^,^'1"!4'

{ t= 1*Lb

, t taen Fa!<.r f c,

A= fnL1

cz

1tIJ

III2 l

J

I - (

b{

x

rJf

;o

bL

);

,A,

LInooSg ,

I *r (t"

l'I 'v

I

e lot l

t^-=

L\ t ' t.t

c ' - C \ ,

=

o#o

f r o gLr l^ 1 o

is -,- |

(\I r lT'

/ , lR fI r tLJ

-1rX; f

l I , b- s)

W{,ys}

X=

3 Xa+ X3s l.

X./1+ X3*?aYr:

L1:l

6. (fS points) Find the LU-factorization of matrix A below and use it to find the solutionvector x of the linear svstem Ax: b, where

ie 1 2-. f . r 3 tlo -G -r

"Tr

s) Q' t = - | '+ E'3r3

lv,f ?

lz 1 21 lrlR: la s s l , b:141.L6 -3 5l Lol

Ia t t If " $, Ilo 0 l l

l*,lq/

o=l'- 'r; l))ld - t s \

l:3

2"7;+ Ys; q

1t * \Yz+79

iIjt

{,I ' r lt /J l

I-rr Pts Score

1 IU

z IJ

t) 20tz+ 20

15

r) IJ

t 100