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Unformatted text preview: Linear Algebra I: MAS4105 (6137) ‘ Second Hour Exam
W. Mitchell 1 Wednesday November 3, 2010 Start your answers on the worksheet; if you need more space, 1. (15 )
continue them on the back. Be sure to show your work to receive 2' (20 ) _—
full credit. _ 3 15 —— You may use an 8 1/2 by 11 one—sided crib sheet as long as you ’ ( ) do not write too small. No calculators are allowed on this test. 4 (15 ) 5. (15 ) 6. (20 ) Tot. (100) 1. (15 points) For each of the following questions, circle either True or False. (a) AB 2 I implies that A and B are invertible. True or (b) Let T be a linear operator on a ﬁnite—dimension vector space . Then for any
ordered bases ,8 and 'y for V, [The is similar to [T]7. w or False (c) The product of two matrices always has rank equal to the lesser of the ranks of the two matrices. , ’ True or 
8/, Ada (3;). (d) A system of 11 linear equations in n unknowns has at most one solution. True or (e) If M E Mnxn(7€) and c E R then det(cA) = cdet(A). True or .‘ﬁg
(100 (c/V ‘ c ” MM) (f) There exists a square matrix with no eigenvalues. or False
o l 1
a; (to o M MIX/R).
(g) There exists a linear operator T with no T—invariant subspace. True or V r [a] Mr W} Far/m/‘MZ‘ 2. (20 points) For each of the following matrices, state whether it is diagonalizable
and, if it is, ﬁnd Q and D so that A = QDQ‘1 where D is diagonal. (Do not ﬁnd Q‘l.
(a) A:(0 0) ye}! ‘D=4) 6?:1'. 3. (15 points) Prove the following theorem: Theorem. Suppose that T is a linear2 operator on the vector space V, and W is a
Tz’nvam’ant subspace of V. Then W is Tninvariant for every positive integer n. F’VVf M W /2c/a&I//‘W M ’7’ 2‘" MI, MWW%7;7 4/ Ma
73/» Mir/M & y/ﬂfé/c 7%» 47/2 3/ AMI///m/ M/gm ﬂﬂﬂv
/A ﬂV/m /97ZIJ 747/ W/j fﬂa/Iq rg/IM mﬁrp H66 76r)éA/é/iz'éaé~c/‘ 9/;1/44’” 11/ 70/ “7(7”[x//EW 5/MW‘W/ 774th f I '
/p1e’ X W 4////fﬂ//’ WA; 7””: ,’;7/4,/M re ﬂag”? “ﬂag/rownoéeﬂz, W W/l f}, /;M/ﬂ/ﬂﬂﬂ’ [j 4. (15 points) Find all solutions to the following system of linear equations: $1+2$2+$3+$4=1 mgfx3+x4=1
I2 / / /
, (0 , , ,/,
.. al .., K’ 0 3 ’/ )
0 /—// / )0 +%>9«J<,=~/ 5*” X15"/ “3/5 ”W
XLr/ +X3 a; 5. (15 points) Prove the following theorem: Theorem. For any square matrix A, the transpose At has the same characteristic
polynomial as A, and hence the same eigenvalues. 64¢ M”) = yéf (Ah—AI)
.— WMZ’AIL‘)
: Wf/A‘aﬁf)
: M(/4"/\I) .—.— 04/64)
13 6. (20 points) Below are three bases for P2 (R). 5 : {1, $5 $2}
”Y = {932,151}
(5 = {1+$,1,$+a:2}.
Find ﬁg]:— 0 &
(a) The matrix for converting from 'y to ,8. (0 ‘
. l 0
. . I I
(b) The matrlx for convertmg from 6 to ﬂ. { 0
0 0
(c) The matrix for converting from 3 to 6. 0 l
/ /
O (7 m
\/
/\
O‘x“
ng
“o
\/ \‘ ...
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 Spring '09
 RUDYAK

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