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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...
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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 mrLLn“. 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 bsiWLP/Hf Wist )
{Laovxu mam—"TO. ‘ IL Hint: Finish by showing thai. almazzwzanm 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“ ‘ijl/ {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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 Spring '10
 MOORE
 Linear Algebra, Algebra

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