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Math 205A Test 2 (50 points)
Name: ~"'S
. Check that you have 7 questions on three pages. . Show all your work to receive full credit for a problem.
- (6 points) Short answers: (Show all the calculations to get the answers. No explanations needed. )
(a) For a 2 x 2 matrix B, det B = -8. Find det 3B.
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IS' scolJ b(f 3J ::::- 9 <c-~) ~ 1- 72.}.
€.A-J- Y\J1.J
(b) Let ill = [ ~ ] and let il2 = [ ~ ]. Then the set B = {ill, il2} is a basis for ~2. Find jJ
suchthat W1B = [ -~].
~ ::::: 8. u'l - ~ \,{2,.
~ ~ =~UJ -1- L}} s fl\i;] \ 0
(c) Suppose 0 is an eigenvalue of a 3 x 5 matrix A and the dimension of the corresponding eigenspace is 2. Find rank A.
t;.ro~ 1~ t-r 0 13 tJ,e... g,}- 4 Sal-J,'w1S
€-'j~"'" ~ = 0 oX Ie.- /fi =-?5,
lhv.$ '1JU\s r~ ~ D -;- Nvl A
So we. 0..~ (\ ivUJ ~ dum tJv.t A=-2-
Rw-k k~ '. ':) =:c ~"" I'tv<lA- -f "..."" ~ A
~ _r6-f_ k A ~ 5".- 2- :::.~...:----
ot- ~L
..
[
]
- (6 points) Let A = 6 2 8. Find a basis for Col A and then state the dimension of
r6 2. 0 N 0 I -0.f:
0 i -2- ' 0 0 ()
B~li ~ ~I ~~ {[il J [~JJ
dJ tV\ Cui A -;:;.1-
- (7 points) Let A = [
10 - ]
. 9 -2. The only eIgenvalue of A is 4. Is A diagonalizable? If so find the matnces P and D so that A = P DP-I If^
.. not, explam why not.
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[
r) - ]
' O J
-- [
G - 41 q -2- 0 I - 9 -(,J
[
b -4 O J
cV [
I -2./3 O
q -b 0 D 0 ()J
~~:: r/~:--] ~ x'-e~3]
B~ IS r- e.oyr =- i e? ]J
-O;WJ A, ~ C7Y1.~ one. ~)-,~~ I~f~ T~.
S".tL we. UV"-..0t- h~ .wo t,,~Ac\ Ihcle..r~
T~I A; rr ~1-- ~oJ.t~.
X I -=.. 2f3 x'1- Xv~.
"
- (8 points) Let 17= [ ~ ] and it = [ ~ ]. Let L = Span{ it}.
(a) Compute the distance from iJ to L.
1\ - 2..
[
3 ] r6/s-
T1 :=:- ( J
-q, Lt
j
li -=- Ii ~ ::: s=- b ==-b- 0
'1 lA'
l~ rl~!:::,"'-- ~ - - -~ -. I
l
b/q- J
[ -I/S-
J
~- i ) :'f)to L 1:. -- j 0 -
[.
2-
]
- C? 2-
L ~ lA \ '2j~ 3/rj
1)\5 t--6J\ t,e ~ -S ~ L ~j)~))
==-~ l ' 7: ~
~ J..L 2/;; f- 4+..i-.^ 2-~ -:::~ ~
(b) Find a vector in L1-.
~ is ff/he r k ~ a.rJ.-. ~u, C<. it- i1 ,>1 t!--.
~ ~:: G~S-J is Ir> 0-. A vtvhJy ~ /5'.
~ 11 his", 2- rf (;j 'l1::=--O.
l~ Gi~
[ t]^ -r~^ ~~ ~+ P,'Jt-D-f (Ac ~/ 6 I C. ~1AJ,-h-.oJ:-
:3 fA of G:--V.
~ a- ~ b~ ri 1 oY ~~ J uY eiJ
i h t vt... G--I"t- > ~.. ~ 1--1- h I , 1 h }'""\ , - d -rY"-0J\ d P cIJj I Of/r ~. ..
- (8 points) Let {VI, V2,V3} be an orthogonal set in }R4.
(a) Is the set {-5VI' 2V2,V3} a linearly independent set? Explain. y [--5"""V))-f L2- C~VJ f C3 \j =- D CD ~ C-~0. Vi> -t Cz- C 2- /" .~ ) f ( 3 ( ~. VI ) = 0' Vj -s C, CV;. VI) +'2~. LD H CJ (6) =- D [S/I/\ <L N I ~ I V;3 If oJ'
- c; ~ (Vi. Vj) -=-v S',:, 11-:::-0' <»-h,°J~9.t J e1-)
S1>;1)""~ 'I~ ~ ~ kb f""h-J; W"jn.,~ 4 hdb.c [ulu 4-.~~DY1 cD) f.A>L ~.t. Cz..=-. trY\ot ~~ ~ ~ c3~" H V1 CL tl,e.- £d' £ - 5" Vi ) 2.. ~/\l3 ~ ~ 0.. d,'n e.o. ..(,., 1'>j~<U'! ck I:- ~. d' ~ S'~C10 ~ f- ~ / 2-V1.-1 ~ ~ fJ ~n '~1 ~l) ~~ stk-.-\s III')UA~ (b) Is the set {- 5VI, 2V2,V3} a basis for }R4? Explain. 0 ct M Irz ~ ::;:.. l;: ~ <>Ad h4JI' 4- IRlr ",,~I:- kA",,- 4 ~ M lb-. 1hL s«- l-s"v). 2.~)VJ3 ~ fY'~ il,~~. HV~ (1;. i:.r Y'-1)~ CL bCAJIS ~ IRlf, ~h-;,.IT ,J;- ,
~ c1f~..cbJ- '
~ ..
