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Unformatted text preview: METU — NCO LINEAR ALGEBRA
MIDTERM EXAM 2 Code IMAT 260 Last Name: Semester ISPRING
Date
Time
Duration :90 min : 13:40 Acad.Year120132014 Name 2
Student # : 210.05.2014 Signature : S ﬁWL‘W \cﬁ \J‘ 5 QUESTIONS ON 4 PAGES
TOTAL 100 POINTS 1.(20pt5) Let T : R3 «a» 132 be the linear transformation
T(a, b, c) = (a + b) + (a + c):c + (b + c)x2. Show that T is an isomorphism. Nate +04. ﬁdmédemufﬂ mt (1+1; ‘3 O
ourc :: 6:; QZG :C :0 UL U
in“: =
He tau: HRCT) :4 03 "Bot M( IR?) =03“ (E) :3. P3 :Dlmasiom Tomi“) &%( M(TD : 3 j {1134. IE“ T TS (in i‘bcwwﬁFLtSM. 2.(10+20pts) Let T : R2 ——> R4 be the linear transformation T(:c, y) = (:0 + 3,1,3; — y, 2m, 2y). (a) Find the matrix representation of T with respect to the pair of basis {(1,0), (0,1)} and
{(—1, »1,0,0), (1,4,0, 0), (0, 0, 1,0), (0, 0, 0, 1)}. :ﬁ 3; (b) Find the matrix representation of T with respect to the pair of basis {(1, 1), (1, —1)} and
{(1,0,0,0), (0, 1,0,0), (0,0, 1,0), (0,0,0, 1)} USING THE BASE CHANGE METHOD from the
matrix you found in part (a). Puti e) : {,(MhibUj owl ,ﬁI:L(LO,GIL)J(oliJOchﬂujanﬂ)!
(O; 019; D) , M33, Basra 0‘9? mvhﬂd) “Q. CW F109,?) ; n I u
.4 1 o C‘ ,1 o 1 I]
._.( .4 O O 6’ 1 1 i,.
o o 1 0 '1 O
o O O I O 9»
1 1 i i\ l C
_ 4 _i \v 1 i.. 2: O i
_ 91 0' ﬂ. 3?.
Q j». ,3, a? 3.{20pts) Let V be the plane deﬁned by the equation x + 23; — z = 0 in the space 3R3. Find
the matrix M(e,e)(P) of the skew projection P : R3 ——> R3 onto the plane V parallel to thevector
(0,0, 1) with respect to the standard basis 6. ' Talc: a base 3m V) $623 iv: CanLO), (Arum)
'TLW f: ( i", ”(11,153) villa {.5 :(0,0,. I) is a lamb 19W— 000 (T5): HUML) (l?) ﬁlth") (T3) “(6%) (ll) Cthcl __.. , 1 
on; mm; P) : mat?) 4.(10pt.9) Suppose that T : V —> V is a linear transformations such that T2 = I. Show that
%(T + I) is a projection. m em a W e: ﬁlT+r) m)
.. .1. [TL +2T+I> :1. (aT4wQI):$(‘r+I) 5.{20pts) Determine whether or not the matrices below represent the same linear transformation
with respect to different pairs of basis. Explain your answer. 1 2 3 1 2 3 j
r __ .. i s
“Reﬁll 2 0) (i 1)  New; C ) 0
1 0 0 0
Nola Heel.— (TE‘. , T€LJ 1—53) is a ell/Wag _mi¢l9e¢0(€ml
verm ) MAX 694% (w T))= 3 Be
53; e... 36': 2 Bill {FJPPJ w— £1I"EI“E;: & ill: 36:.)
ml is, clm (mtg?) =4.
\X/umu. T i S. ...
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 Spring '10
 uguz
 Linear Algebra, Algebra, Linear map, linear transformation

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