m526_s10_pract_t2_solkey

# m526_s10_pract_t2_solkey - ~lATH 5 26 P RACTICE T EST 2 M...

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~lATH 526 PRACTICE TEST 2 MAltCH 15, 2010 [ () 7 -1] 1 1 . pI" 1. (25 points) Let A = 3 is -9 9 ~7 · / O O l [' r o I o \ ! \ 00 I.J _ ,., () {r-l o ~ , () 0 (~ I 0 \ 1 - -_ 0 .... - () - \ -- :? -2 L f o 1',oJ [ 1/ L ~ 6 7 - J, _. = o 0 2 (1)) Usc the L - U - P factorizatioll uf A t.o solve r..:.[ .~~. ~ ~1] [::'.~] = [~~ ]~.t 1.) -9 9 .L':3 33 L (U)?) ~ f{ ~ --.> r 2 c: r: ,,~ /' r: z: ~~-~ .:> --I - "«'J-+- '- 3 -" / ...:.-;,"';> 2. o () 2 .. _ L)-:~) 1,-. (a) Find the L -- U -- P factorization of A. 3 III 2>( ..... -=-( o . '

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pr.2 (10 pts.) Let A be an nxm matrix, and b - a nonzero vector. For each of the following answer True or False. (a) The set of all vectors x = [ ~: ] with x, + 3." + 5.'" ~ 0 is a vector space :r l ] (b) The set of all vectors x = :r2 [ ·<:::3 is a vector space. (e) The set of all linear combinations of the columns of A is a vector space. (d) The null space of the matrix A is a vector space. Trl/2 (e) The set of all 3x3 matrices is a vector space. 82 pr.3 (25 pts.) Let A = [; 1 =i ~]. }<2 - 1 K( (a) Find an echelon form of A. L .. 7) -'1 :] 0 o o
r- .: (b) Find the reduced echelon form of A. J () 0 -2 I ,-_ ..... (c) Which are the pivot (basic) variables? Which are the free variables'? (d) Find a basis for the column space. I ') P ( r: , ["'--7) [[;]7 j l 6 (e) Find a basis for the null space. . I (' ,1'-- X -r- r''i - )V-, .:::,: 1-' I ''7 <...J - " ,~. kl ,:.()

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