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Math 205B Name (^) Final Exam page 1. 12/15/ -'~ "
[
- Let B = :
]
~I(R)~= ;Jvl(!5T)~Nu(U -: 1))= Nv{[}
( r ,0./ -", [
[
,r/~27l
a bc.r" j), tr...;/a,t01'1=1 i, II -Yj) (IB) Label the row vectors of Bas rl, r2, and r3. It's clear that {r2, r3} form a LI set. But rl is a LC of r2 and r3. Find that LC using appropriate techniques. Then verify that it's right (writing your vectors
horizontally)by direct evaluationof your LC. "- , ~
We. ,.,.w Iv Iv c< 2 f ~r vlJ/d i r; = O{('.... (3~ , "", ( " ~ / 1/) : ~ ( 'i I -2 I 7.) of r ( 2, J) <j)) or
e!] ~ ~[-]J + rU], ic-rP4"aj.m1fax~l-l
>0« ~-i/>fho.I rJ~o/5f.. u ckt(: ~
If -2 1.. )
1' (
)
(
-.Ii? ~ (^) , -.. )
+- (
?21 r. ~) ~ (
J. 1>'55"' /S t) is I I ~ -; s s S, 5* J .sJ S" , ~ , (^5) )
== (I, 3, lJ) fAo bt'rr?1 e
2. The points (-2,29), (1,8), (4,23) and (7,74) all belong to a parabola of the form f323? + /31X+ {30.
Find /32,/31and /30by setting up and solving an appropriate system of linear equations. Show all work and any matrices and their rrefe;that you use. (This is not a best-fit problem!)
)/!)Ce.(-2,Z') ~ {}YIIu /MiUI..J (,It!hAt'(; 'if7- -Z~, T ~o ::: 21. ~inlA/J)
. pz. --I r, t fo -::: g
-£,. oIL I/,«~r'Ali J"'" ~ /6f" if, 'po'" 23
, '11 f' + 1,. T rq -= ~y
-r( O;(rtoffYn"', f4VjrNwfd;ntJrJK &4"/~-~
J ,
fro 'i 'f' r (^) , .J
(1 := Z. f1 = -5"' n := II cm/~, ~ -t atraiok
-~ -, lOr----
'- 2x2_~x + II
(IA) Find a basis for Col(B).L.
0 S/:])
: V:7J);
3
/ (^) ~ ~
I 0
/
-'
1
[ ] rJ 0 ( 71/> OJ J/ 0 o. ()
2 'f
I I
. I 0 0 g ('oJ b ( C> 23 () 0 I t'f D 0 0
2.. -S' II 0
fD ,
" Math 205B Name (^) Final Exam page 2. (^) 12/15/
- No single parabola of the form /32x2 + /3IX+ /30can contain all the points (-2,29), (1,8), (4, -13)
and (7,74); you do not have to prove this. However: Find the best fit (least squares) parabola of the form f32x2 + PIX + Po which passes through (-2,29), (1,8); (4, -13) and (7,14). Express your answer in decimal':!. ( 4 -2 I / ' ]
~o WG till: 7lJL e..;ki, I ( I ~:I- = 8 ) td/f,fLYIShint If we JrJ" id If::, 'f I r' -/3 f} I
'11 "---- r I '~o J'f' ~ CtJr~r? /eMt .s'b~ cvh..
x: tfu CrJrr-fYldjMjtn~ ,n4/tlk ;
1"0 I 3 S- ~
U [
I 0 D
J ( (fjJ(ld I e.rrcr /0 '1/(P rv 0 I 0 -II. 2 ~ /p~a. '1/ '111 0 () I £,2~ x/o -'2- k" -@ ,..:!""" :::: 0 i:I /l;t u.( , So t /u.,i~.-, f'akk h /1 X2- -/1. LX 1 1tA~ 0) lD~1twi Wt. SoJilt: X".xr =X'j 2(; '1'1 LfOO L-fDD rO. to / ~ ~ r :: !:J
- Even though the parabola in problem (2) contains three out of the four points (-2,29), (1,8), (4, -13)
and (7,74), it is not the best fit parabola. Show why it isn't by computing the sum of the squares (SOS) of the residual")it produces and the SOS for the best fit parabola in (3) and comparing them. 1l orjind ff1r,W" ,'n (2.) ClJIlf1v~J (-2,2'1),(t) 8 J, ('I, ;;)" c",./(~ 7Y) X> ~ ()r>t" Oft'- 1?1:1t)le1AO ~/~J! 2.3 - - /] = 1C> ; iItst>~ t:. 121'. , J -- tJ- -- ) (a!rtrnMcl...:for Y; 2..1- 5 X "", t!. r-eJd« v.;... ;1<J Cl»r.&, .f() ~J ~J' ~ ~.. < i
Z
~!
Z
]! O J ,t?. ) t. ~fI$(JgdvttJvr" ,; ~ (^) f riil- ()IoS , ~ ,8 - i ~ 0 M;L JOS/ . 11v. ~lIv"
' z.~ -13 3' 'It. 'I. t..
c 7'1 J'1 ,() 0 0 -tD .dt. -t-() =/2'1"
fiyr :r:Ii< ""-//.2)(, 4..- flldtc/G . ;/ v~ A.< J'~ Iy xH~ = f
- !~t
]
l 3'i' J (^) [ 2.Q J (^) [
S:Y
~ e,g;~ ' ,dvAL 0 - %/2.. % - -Jf,. 'Z. Y1A}N t (t4 ~ 3.1.. - -0 - /(, t;<t. ~ '} 'i - 5".'1. '^ a..i^ IL,^ So..c-j^ /tv..^ ft4;b~^ r..:J'ZCU S".'i"to -+~(P.z.)'l. -+(/".2)1 +(s.~7.. :;:::.581.'2-. JIk It c; sfI1A 1L. 1276J ~ flt~'QU. ~ t\ ~{fv 4 -Jt~ ~ klYhwfIIN! ~ r~ (2.).
,Math 205B Name (^) Final Exam page 4. (^) 12/15/09 --, ;. ~ 5E) Find the corresponding change of basis matrix P from N to M.
~ {1m/)
[~
2- 1 0
~J
5F) Express Z4in "[ ]N" notation.
r~ Iu. f(Rf:F r /<t- It ir jl;Ab. ~/l~ci~(1dJ<- )
~ ~ve. t~ ~ -'1~, -t t17.. - 5"2.J ) SO
IIJ ~
[
~;
';f -5J
5G) Use P and the answer to (5F) to write Z4 as a LC of the basis M elements.
[l~JA = ~[~n~ t~]
5H) Compute that LC using the appropriate matrix product to verify its correctness. (What is that product?)
3~~ rl~J =: [~~
]
= 23;' -tIlt.,. - /'laS" J ~~I^ dJ
. L /44. -/
JL f£~ [a, i!b ~>] ~::] ~~~ I~~
~ 0 ~rrec;t
"2~ =
'
-
" ,Math 205B Name (^) Final Exam page 5. 12/15/ ;.
6. Let A =
-1 -5 3 9 I
-48 -40 (^24 92). its rref i" 0 1 -3/ 94 10 -42 - I
Here are two facts: (^) 1) A':"-4 is an eigenvalue of A.
2)u = [~] is au eigeuvector fur A.
6A) Find a basis for Nul(A) (see the rref above!)
~- I
0
1z.
11/
J ,
000- o.. 2- I
6C) What is the eigenvalue for the eigenvectoru?
n~ A[t]=H]=s[iJ)~ktJ A=S-~~ed~v~.
6B) Find a basis for the eigenspace.for A = 4.
~t?lt ~ t. ~/) "f': ntJI ( IJ - '1 I 'f )
-. /II
"".r ->' ~
- f)V -'fg -"iLl z"\ '12./1:= f1U] Q'1 tD - .f~ -/(,&, -'1¥ -'to Z'f g. 8 I
6D) Show that A can be diagonalized by finding the appropriate P, D and P-l.
(fl)tYI fP<~ ell i A~0 w ~ ruy'",~ jA l/)( C wrdj 2 tI~/tP,ttf.:!J~r-'
~ ~{3 f?(;c) t.J<. ~ /wJ f'h1fU-.t&~ ~ ~ JP!Iit;rejtmcr em k cl4~r,.; c,vt rw~t i [{,/t ~ ~. r.i..~L t A~ '1] .!tine Il h'ljX 'I... ~Jh r!; ~:I ~J
VX- t.- 0 = r 0 ~-:> - ~-:~<;
'-I'
S
~ ~I ~ ~I ~ /;r Ik
/)J MO~ p=
0 Yt. J/-$' 1')/» ( ()
-^0 I
~ ~ '- h),? ~v'/~~ ~~ ~~
~ wP; ~v ~ 1 D. ~ ~ ~~vlt~ /f17V uP. /;. It- ~~ ~t14I./.kA I /1) p,;.ff) f»
!~~~. t
~". <>-
6E) Find the characteristic polynomial of A.
0~0-i)0 -s-)
.. ,^ I Math 205B Name (^) Final Exam page 7. (^) 12/15/ ~.,
- Let H be the sllbspace of F consisting of all flIDctions f in F whose graph intersects (ie, crosses or
"touches") the x-axis in exactly two places. Also, let the "zero function" be a member of H. Explain why H passes or fails each of the individual parts of the definition for H to be a subspace of F.-> D II.
I{
, I ~ @ 0 f. H
I\tl.~. IJIC.St)o$ru- '7 @ u.. (^) ,V (; I( ~ a,+-e- .. f:1I
/I
(J) Stll</it (II ~ sa t.
& "- H j -!f,/j rlor.t... : 0). D dl heCMlk. ~ dMI dtl ~f;.
. ~ 2. ~ lc.l
0 ~Il~ ~ tOf)5/~ a .= X -I ~ ~:: - X. -r {. -;t Jrj a :~/m4tdJ. It X-MIS M. jf'(f. X -:1 a...(X=-:1; tL Jrr!j v J?~ ai. J5i x~ z. 4-f x= - 2. ~J -> u.-:> a.-I c? /d ":1 Iv I/. lIaR/'VI).. l!-+V= (x-&J)+(-x2.r;)::::(tt{~) -,L
. dvt!t !iJ- £d:;; ph /I k d It~r ;/1#~ ~ X-a",:' Q,
Jw() Ilia. /VtJ~ iJ It. JG'W ~k. (JJ f;9~: tJ; UJ: fl. W: sdf k a ~ ~.. {fl<. :L: -tL Jrfj "- ~;'ff~ £ )(-~f /;f:C/1 ~~d; ~ ~Xl. J) ~. 14-_ 1. It - .~- ~(S'a., . 1h(...,su. : Ml[/.>&U,$7hC X-Am c>t? \ J/ ~ t S~...c Iz.v X-CpoM/.1A/1'>XI !( Xl. l ~to. If .f: 0 Ill! oil-:: 0 e II ~ II~ SP//~.J:t. Ovx :Jt: 7f 1l fa tt. ~ :k~ I /:.. ~;}~ {5 db ~ ¥? Ii ;, I(
