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Unformatted text preview: Question 1: Let
A = C . (a ) In A). {1~~
,rll ~ !3l!~J\1 ~=~  Y~
of the matrix A (d
0 t % t ~ ~ 12
/j.. 1/ . ompute the LU factorization ,lj 1'6 ;. 2/. ;;. IL
/'iJl t no swap rows /3:.. (b) Compute det A. 7'10  ~~ ~ ;lIto::;' Y3 L:::. laD Y2' 0 ~ j~ \ Question 2: Consider the linear transformation
defined by T :
R3 ~
R4 (a) Find the standard matrix associated with T (i. e. a matrix A such that T(x) = Ax)). (b) Is T onetoone? Is T onto? (Hint: look at A.) , ~~ r
0
I +
I
I ~~[nX~r
t
\. 0 (\~t, 6 .. , I o I ,. I oJ T evJ:o ~ ~ ~ ~ ~ 'r'~J ~ L. 'fI\';'t ot Question 3: Consider the followi , owmg matnx: A = [ 1 1 2 2]
'po 1 2 3 3 ,;1.\ 1 0 1 0 AJ (a) Find bases for the column (b) What is the dimension of c~f~~e and the null space of A. the columns of A form a b is f the columns of A span asis or R ? Explain,
R3? ' D 0 \ 0 \ () o , '\ 0 o
~~ X1.. 00 , o
\ t 'to \ \
o
\ o o .
IS Question 4 Sup pose V IS a vector space and B = {b b 3 4 . . . a basis for V. Consider the follo wmg vectors: . . 1 2, b , b }
~~
VI = b2b1 +2bb2 + b3,
1 2  b3, V3 = 3b2 + 3b3 (a) Find . . (b) Determine .whether {v 1, V2,V3} IS a linearly independent set b f , : a asis orH=span{vl,v2,V3}. Whatisitsdimensio~? V2 = + .s _~~ ..I)I.)~"
\J
jIfY""""""r
\.IV \i)1 .;' 1'clJ l
\ . b, \ Z tb.,.. '2I _ r'b t ., 3 b \ 2> UJ;l3~ 1
I o Question 5: Let lP2 be he ~ ace 0_ polynomials of degree at most 2 and consider the linear tr 0_ aion T : lP2 ~ lP2 where T(p) = p' (ne derivative of p).
(a) Find the image throuz To: each of the elements in the standard basis of lP2 B = {l. t. {~}. (b) Show that for any poly omial p(t) = ao + alt + a2t2, we have [Tlp)IB
where =A [pIa, [f']13:. A = [~ ~ 000 ~l
l
I
::
o 0 0
Q'L [t1 ~ ~ Otl) Z,t) 0 } 1 )1, 2t~ 0.., A, [p J '" ~ [~ ~ ~ [aG..,o
....., V' ~
(A, }lp) J ::  i
t t2
'1 ...
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This note was uploaded on 02/09/2010 for the course M 340L taught by Professor Pavlovic during the Fall '08 term at University of Texas at Austin.
 Fall '08
 PAVLOVIC

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