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Math 205 Section B
Final Exam (85 points)
Name: S~Dl~u~J
. Check that you have 9 questions on four pages. . Show all your work to receive full credit for a problem.
[
1 -2 1
]
- (10 points) Let A = ~ -~ ~. Use this matrix to answer the following questions: -3 5 -.
(a) Find a basis for Col A. What is dim Col A?
A rv
[
I 0 0
P, V) +J IV)
B C..$ 15
')
0 i^1
I~ yJ 1 l-w"'O to I \A VY)V.S. r. ' 1
, ri
]
'
[
]
( tv f ~\ A:::
. L.
?. - it 1 0 I j! .--3 5- J '
0 I
cV M G.) I/~:::: 2-
(b) Write one of the columns of A as a linear combination of the other two columns. (You can use your answer in part (a) to decide which column to write as a linear combination
of the other two.). N'" ~ 0
.
3
..
"
.
\
<
.
H ert
...
" tNrJ
...
c'th Vv ..
<t:t) ~\j' t~-
.
e..... A e
1
ell t, V
l
0 0 () ~P, _1Y_^ ~~.^ ~ (~fV.,(,,--J-YI(- 'f-.^ I~ yD^ t,:,^ e.-^.
[
1 -2...
]
0 () C S\jJ -~ Bx= h/"'-I~€Y~ /)~ 's
.
.- - ~ J ~ "
c; C],^ (2. J^ I ~-^5
[
-'
",,'" h ~-! )
-Th
~yV( J C3^ -=- '3 Cj t-^ (/-
(c) Is the equation Ax = b consistent for every choice of b? Explain. S. (^) I Y'\Ct- -1--\e...~t-\Jr'l(( I I.^ d~ e ~Ir, '"^ e to f'\ hJ'! I;^ ()oI 0 ~ Q^ 1\ C(11(:'fl.^ J Y(H- i
G"- P IvO b- in eA J.. I)-~, l"h e.. e ..,, (^) /Altv.Jn"Y)^ A X^ -:::^ t;; /). (.
)".0t. (Ai l\Ji J ~-{1'\ ~ --tv," e v( tj C {'\OIt,C. o! b-
~(~ VL
u.
- (10 points) Let Pl(t) = 1 - t2, P2(t) = t - t2, P3(t) = 2 - 2t + t2.
(a) Use coordinate vectors to show that B ={Pl, P2, P3} is a basis for JP>2'Find the polynomial
if in 11'" given that [q]B ~ [ -~].
~ f"'l ~ fO' ]
.- ['2-
tl =: L-~ r p 1- =: L-I I P3 ~ rt J
r
I 0 '
l [
I 0 0
]
,. ' 0 I -2 IV 0 I 0. So !p:, is
--I '-1 i 0 0 I
-. --, .-- J
[
I
]
r 0
] (
] (^) [
"
~;:: I PI - 2 p,- f' PJ> -= -~ -'2-l-: + -f :::-'L ]
l\IV.I' .~. tz,~ S"~ _c_ ris" Yet h~ jjJ
_l_ b) .t1-j ~ i lrl-)
{~, bCtS (j~ y IPl-.
. 1'0 fW=~-~H2-t'2-
(b) Define a linear transformation T : JP>2- }R2by T(ao + alt + a2t2) = [
ao ]
. Find al + a T(pd, T(ih), and T(P3)'
liB);:: [o~J:oh:J
<Ty)~)=-
[
a.
J
=. [
[)
J
'~
I 11'(-1)^0
-1 Cf~)=- f (^2).
.
[
~' L-2-t1 .] (c) For the linear transformation T defined in part (b), the kernel of T is a subspace of JP> of dimension one. Find a basis for the kernel of T. (Your answer to part (b) can help you to find a basis.) Is T one-to-one? Explain.
W~ k,w TLfJ-)= [g] (f..YIA 4,)).
So ~ ;5' If\ k~Yht' O~ T.. .-. -
~jrw.. ker T IJ O~ .tln O'Y\€.. I 10 ASll h,y ~U T - t f>- S.
'I h<- L1lA ,,-.h1M I ( 'i) -::: D ~td 0.. N:J ,,: h'Vt "I [0 k,h ''>
bLL ~,'-lJ <. p;. LS C/Y\ e .\c l-J, v~ ()~. ~ is t-7 l/~" C;''Y).
So '1" '{J -1\0 l- 0 f\ t.- h, - OY\t..
4, (12 points) Let W be the subspace spanned by the two vectors '" = [ j ] and "" = [ -~],
(a) Is {UI) U2} an orthogonal basis for W? Explain.
4,. U" = b- 3 - 3 =-0 So ~ 4, );:j~ J if ~f, 07{l,°YA~r Jek '
t:" ~.-( 'R. J" "I.": t-J el-, if II """yl d i I'. J "f €A-J u--.f.
A~ 0 ,tht Jet {l4 I ~ 1-~ JfO\ ~j' W, ,
So {4; I Ih 3 i.J t\ ks',s {;,-r Wlhld It U. <", o-rfhcgoV"j hi1ftJ
(b) Let ii = [~], Find a vector in W that is closest to ii and then find the distance
between iJ and W.
VtJ".r~1- W ~,J: - l!- clvJeJ~ h Y U ~""'1 b(j .!). U ' ( - It (^) (
j '::~' v~ Lt( 1- ~(L J.- ~}... ~ .v,,= 0 4, .IT, ~ .u). g-, U~ ;:; ~t 3--:;;-- (1 -
~ :LU; ::: LTi) ~ [i~J\h'u >-::; 36 -thC( = 5Zt
" 51-t^ (;^ L;^ 1'2-' J ~ D l\
'D,S hWL "',,~ 5 <1'\ <\ h):: ellS} Vif\ <t b~ '1 £1'1'" 'j ,
, ='II~-~Ii ~ 11 GftJ^ I^ :::vor-£i~ = [~. (c) Fmd a vector in W..L,.
V
i, .L
eC{) 7' i Y\ W :::-
-. 'j - Y
[
' I/1- 1/1- (^) J '
(d) Find a basis and dimension of W..L. (Hint: Think geometrically and use your answer to part (c),) , tj i U J- U ~ bcU' i-e ~ , ~-J [e c"v l'n (,-J - 2- ' "'I f ~I).- ],,, ,~ hJ u Cl J'u. bJVltC€., (;1- eli mU\Ji~'h 2- 11', IIZ - .so
"
- ' IIVb. ~~.L iIfheA CL U he. f
' t TPC~l'Lvlq T
f t'\ n ~ In If-- ' I blc.Y\e. 111 IIZ3, So (^). ct ry\ vI-~ I. .~ - 1 I 'Jl_.I ~
I a Mn -l if"t-r iY) Vv lAir B teJi 1= t [,lfd J liAl.~ (,HLC L h).1- ,
Il~ if t(.
-f.c ~ ,j
BttSif (;Y, ) [
~i~1-1C
<4j,U)SfCA {.( - L I Jj "
~ { [~lE --4 '-t
(^1) ~
i 0 1/2-
] {^ l
--i/2.
]
{ ~ r~ ]
?
- ~ N 0 I - I Bet-n f ::: I j ~ L.l ~ J.
b 2- 0 0 0 I
\'--r A ~.I '. A - 1-::;.
[
~ -~ ~ ' J
rV I ~ f il ' y,cu/1 ~ i ~t] 1
. 4 D ~ La 0 () J .'
[
2. -,I -
1.
.
'n fh 4 (l\J, ~IU»1I1.=J4'f~tl ~ 3 ~o f=-
[
2fJ --'!
~I 2 '-L , p
"\ ,;i-t ~(.h. Gvl(,(lIv-f\ b'1 3 ~ 'rf~CL. ~'/3 '2-l
- L. ~1 ~ H) (j '2-. 2/ I C'.O--t-h (,du ~Y) rI- leu"" I. I ~ 3 (c) Write the new quadratic form with no cross-product term. Is Q positive definite, negative
definite or indefinite? Explain..
U J€- U:f'" vl,-t,u.,d 01- A h, ~U-7)I-t/h c h 1'\ J~"""l.
Ct ~cJ y ~hL ~-rfh:' I '?J x/I- -t- 7 )(~ -t x~.
S i/\ lL 'c 1\ 1t,e. ~1r'v<:~l..1es d A G\ ~ rUJi h'vt ,
Q ! P on h ~ d _eJ,) f_ J{:.
- (10 points) A quadratic form on ]R3is given by the formula Q(if) = 7xi - 8XIX2 + 8XIX3 +
5x~ + 9x~.
(a) Find the symmetric matrix A of the quadratic form.
A~ -'-i lJ'
0
'-t
7
-~
Ii
(b) Find a matrix P such that the change of variable if = Pii transforms the quadratic form
into one with no cross-product term. (You may use the fact that the eigenvalues of A are 13, 7, and 1.
~i A == J 3 ~ 4- I!> I :::;:. [
'- b ~ 4
J
N [
i
4 -~ 0 0 'i 0 -'-1 0
0 ..-1 °
J
\ 1/2-
~y ):=-1 ". A -11=-
t
o -~ 4
~1-/
1
iJ!J
- (7 points) Suppose A is a 4 x 6 matrix.
(a) Find k such that Col A is a subspace of JRk.
k:::- '-1, (b) Is it possible that dim Nul A = 1? Explain. (Hint: use the rank theorem and your answer in part (a).)
&j ~"- Ya" l -rr" 0 rtAV; ,
\ f'\ CL
(;;::;. Yd~l A -r ctyv) N0v~ A
CO! 1+ Lr C~ Iu. [, S fC' '-t 4- if \ 1'0" l
Nvl A
A '(.1 ct.; 1Y')(;i~- '-I
Jo
~{),-_r (^) (;...t:)Y) 1:1. j' t (JY).0-. c-vt^ (€.O-J b^
2-
So dJM^ J'...'vJ^ A^ -:t^ I,
- (14 points) Short answers: (No explanations needed. Simply write your answers. If you do some calculation to get the answer, show the calculation.)
(a) If 4 is an eigenvalue of a matrix A, then what is det (A - 41)?
J~. C A- ~T) '= o.
(b) Suppose T : JR4 ~ JR3 is an ~ linear transformation. What is the dimension of range ofT? ~
L~ M I'"'Ct {(j e.: T -:::. 3 (c) For an .?rthogonal matrix<,;;.- U, what is the inverse of U? .T U
(d) Suppose B is a 5 x 5 matrix with det B = 10. What is det 2B?
dt1 2-~ ':=. 2' (,~el- b ':::: 2':/ J 0 -:::::3 '.
(e) Let T : JR3 --+ JR2 be the linear transformation given by T(XI, X2, X3) = (Xl - 5X2, 2X3).
What is the standard matrix of T?
[-T( t1 ) T(^ r;.)^ I^ (€3fJ
~[~
~iJ D ~J
(f) What is the length of the vector [ j ]?
~
~ .~ - 1- '1-. ').. - ~ I to -f (-f{-IJ -- J
[
] [
]
(g) Let A = PDP where P = 3 3 0 4 and D = 0 0 4 0. Wnte 2034 0005 the eigenvalues of A and a basis for the eigenspace of each eigenvalue.
b(tif'vU.lt. u of It: 2- J 4 IS-.
B (tflf A; -r eA<r'" Jf c\ t:t ~.
A::: 2- ,A :::. 1
[[-tJt fl--! 1i
A~~
{ [;f1J.
(h) What is the characteristic polynomial of the matrix [~ - ~ ]?
(J-~) (j--(l) + I
::=- ).'2. -- g A 'f" \ b
&tf [3~A
-I.
5-A J ~
