M 340L-Portilheiro-Fall 2003-Test 2

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

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

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.

Page1 / 5

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

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online