This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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. ...
View
Full Document
 Winter '10
 uguz
 Linear Algebra, Algebra, Linear map, linear transformation

Click to edit the document details