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Midterm 2 13-14 Spring.pdf

# Midterm 2 13-14 Spring.pdf - METU NCO LINEAR ALGEBRA...

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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.Year12013-2014 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). Put-i e) : {,(Mhib-Uj owl ,ﬁI:L(-LO,GIL)J(oliJOchﬂujanﬂ)! (O; 019; D) , M33, Basra 0‘9? mvh-ﬂd) “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: Can-LO), (Arum) 'TLW f: ( i", ”(11,153) villa {.5 :(0,0,. I) is a lamb- 19W— 000 (T5): HUM-L) (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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