108APracticeQuizH - Name#KJEAI—m J Mathematics 108A...

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Unformatted text preview: Name: #KJEAI—m J Mathematics 108A: Practice Quiz H May 12, 2010 Professor 3. Douglas Mooye 1' Let 733%) dEflOte the space of poéynomials of degree thee, with basis 7 = (PO‘PhP2aP3); Where 330(3):..1‘ 132(3) m 5'3, Feb) = 5132, 193(3) 2 2:30 Suppose that T : 733(R) ——w 393(k) is the finear transformation defined by 2 T<p(x)> = 525(3) + 2§§m — 3pm. What is the matrix [T]; of T with respect to this basis? E... .- IO 5 :— (Pa 9; P2» P3”! 0 E0 \ £2, , - ‘2: !-3 ‘T : eager-(Foemiw‘o L 1‘ \Q g; 4. 2, +41~3m :' (FDP‘ P2 {’31 “3 O 2 a. Let "r 2 {vth} be the basés for R2 defined by Ma), Ma) If T : R2 -+ R2 is the linear map defined by T ((2)) == (3 I?) (32:) What is the matrix [T]; of T with Iespect to this basis? b, Suppose that ,8 : (8;,62), What is Q M [Hg 2 (fig 0 (9%)“? What is Q‘1=if}'é? c. If A = (i :3) , what is Q‘IAQ? Q"‘AQ=- ($2) l0 0""! ) 3, a. Complete the foiéowing sentence: A fist of vectors (v1,v2, ‘ . qvn) in V is linearly independent if and orfly if . , \ -=' .4, ~z- “"3" a" V! “I" 9.»: V2,. “I” , , , 4. RF VI”. t: O 3::- El 1 :: A}; -"'-‘- ~ ' * —-- f," H :7: O . 13‘ Suppose that T : V ——r W is an injective Linea: map, anti that (vhvz, . ,vn) is a. Einearly independent list of vectors in V. Prove that (T(v;), T(v2), . . . ,T(vn)) is a lineariy inéependent list of vectors in W ‘ Hint: Stat by assuming a1T(v1) -i- r ~ -+ anT{vn) 2: O, wa— ‘—-7 my tam/m T [mivi ”V “' 4’ 5",th j 3: 0 .q- "‘7' - {in CkVt4‘H' +mm‘x/n E: NLF]. wry . , "- ___ Z. 18.4mm) T W mrLL-n“. 0 -~- mm, ) N\ 1 ] M {ED 1 ' ".3,- my 7x1zmm, (MU: + . + anvn =2 0 . "- a—«v ‘ j/A- _, (ff?) A'MMCHEJ {V22 ‘3 ‘ " J Vn V) «by b-siWLP/Hf Wist ) {La-ovxu mam—"TO. ‘ IL Hint: Finish by showing thai. almazz-wzanm 4. a, Complete the foiiowing sen‘aence: A list; of vectors (vhvm. . 4v“) in V spans the vectox space V if and only if fl V , M4:- ry -—-'? "“7 ‘ "C; 6 V W V :3 0"ij 'l” a. V?“ "f“ II, + 91“)“. V“ ‘ij-l/ {3797310 ad} 962' 1 1"} ans F: b‘ Suppose that T : V ——> W is an surjective linear map, and that {V3 , vm. . . ,vn) is a list of vectors in V which spans V. Prove that (T{V1),T(V2),‘ ‘ . ,T(vn)) spans WT . ' ' _f: ‘ 33“,};th {I}? r \,J , £1ng T 4m) Ora—'33:“, .Tir'hryuu waits} fill/‘01.) (,Vl ,‘ Va 3 I I 1 3 VM ) gag ‘fj .1 1 "a, _..:;,. ‘ 0 mg ‘Lfl ' "if ELY \Eil‘) V :3 on V] 4 “’2. TL 23. .7: E“ {If} Swrl’JLJ l1 1. 9.13» ’ ~ ‘ J n x w? ”—7 — "3" f” *y") CHM»; "V7 : "TLV) 3 “T (,(l‘Vs "4r 6.1V; 4' "' "‘9' Wu “H ...
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