M 340L-Portilheiro-Fall 2003-Test 2

# M 340L-Portilheiro-Fall 2003-Test 2 - Question 1: Let A = C...

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Unformatted text preview: Question 1: Let A = C . (a ) In A). {-1~-~ ,rl-l -~ !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 one-to-one? 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_ a-ion 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, 2-t~ 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.

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M 340L-Portilheiro-Fall 2003-Test 2 - Question 1: Let A = C...

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