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Midterm 2 15-16 Spring.pdf - METU — NCC LINEAR ALGEBRA...

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Unformatted text preview: METU — NCC LINEAR ALGEBRA MIDTERM 2 Code 1 MAT 260 Last Name: AcadYear! 2013312016 Name 3 Student No.: Semester 2 Spring Department; E SGCthH: Date 222.05.2016 Signature : Time ”9-40 5 QUESTIONS ON 5 PAGES Duration 3 120 mm TOTAL 100 POINTS Show your work and justify all your answers! 1.(18pt3) Find the solution set to the non-homogeneous linear system in R4. m+2y+72~3w=10 ~—:c—y+z+7w=9 3x—y+2228 tail-”5:10 l4?*3:lo 1 g ,«u Kr?.“l"l1 31-13 ~ 613‘1113 {+31 3 ., t 013 O “‘1 5‘ .1! {35 Rum 3 i a 1 "Stio fi'afl‘ { O -3 Hip—.23 Rmfif, 0139119 ~ 0139;131:1495“. I 003131!!!” 001113 ’1 O 0 "-1:'-| x—-&w=-1 N 01 0 “‘1 Ii”; .7:> 3...?“3‘5- DO 1 1 '5 2+»: 3 50% ==l “W, 3:“; #5le =~ 51°in (an—1,0} and g: (”Ln-5‘, 3,0) is a smué ”aim. 1L0 nan-A07 Sea 2 S‘pa-«lHflMJU + (-1,—533, 0)‘ 2.(9+15=24pt3) Let T : R2 m» 1R3, T(m,y) = (a: + y,:r m 31,32 — 2y) be a lin— ear transformation. Fix the ordered bases E = {(21 = (1,0),02 = (0,1)}, 13' u {m = (1,1)”;2 = (1, m1)} of R2 and F = {f1 = (1,0,0),f2 = (0,1,0),f3 = (0,0,1)}, F’ "—* {ul = (1, 1, 0),u2 = (1,0, 1),us = (0,1,1)} of 13.3, respectively. (21) Write down the matrix M f; (T) representing T relative to the standard bases. : ._ _ ,1_ 1 1 ‘1 (1)) Write the matrix M‘gKT) representing T relative to {E’,F’} USING BASE CHANGE METHOD. Hint: find the base change matrices P, Q so that M I 2' (T) = P-M§(T)-Q. . P ' Q n u ’ F’ I F E MiCT) 1-“ F' (1,5) ' mg (1’) ' m 11/11.) H H N Ill. “1.. -VL- 1 1 1 1 N ame: ID: 3. (15+3318pts) Let T : 793(1R) ~> IR“ be the linear transformation defined as T(:v(w)) -—- Wm] p<w1dsc,p”(on. (11) Find L01 (T) Im(T) )and their dimensions d11n(k01(T)), dim(Im(T)). F1151“ flak M [Girl-43;! 4—13,): ‘+%x3)= (Qt—1a I+I+ 32134»? 10.3) Haw. MUG-WT) TH C1,, “as-:0 ‘10: +5???“ 0 TL Wham-“wi— f’pc): 01‘4Q1X3*a(1:9x3) .LlyLrs nan—(T): Speai 1-41 xi) =>6¢n-5m67fl24 Batu—c! an DIWJM gar-MA we. W 44-4.. mfimCTJ) :2 414:3 => mm): 1123 (b) Is T an isomorphism? N 0 ) mtfl % 4 O) 02.. 1114193021) =— 9 #3 MM OK”). 4. (10+5=15pts) Let. W = {:L‘ + 39+ 2 + w = 0} C R", v = (1,0, 0,0). (a) Find the skew projection P in R4 onto W along the vector v. m a w: fizmmv) ,rzrss«,+.o}, é3/OJ0,¢*IJ 2w w «ml pw4' #1.: (wow. 7k F‘s/A444) t. a [flash for 12" and ’P[%~)=:£a,1.s££3 mam? NoL‘v M etlfi't)e1_=£t‘£l 233-: £’£"£Lj 6,316.4:{14 (b) Compute P(1,l,i,1), {DI/1,4,1”):- (“S/LU) 5.(10+15m:‘35pt3) (a) Let; T : 1R4 —+ IR", T(3;,'y, 2,111) = (z,y,w,3:) be a linear transforma- tion. 13 T a cyclic transformation? Justify your answer. Tm ~=anI-~)G~IR". To. T534339”) 141V): (“03, 33%)) T3[v)c()t,z?-,v):v_ So) S r"- AA“ Spm}V,T\-r, 1%,...jc- air-53V, ho T‘vj £3, mi) Manna-Le. 01 3‘34;ch M- LT. Hm; To M+ agebc. (b) Let T : ’PaUR) ——> 793(R), T(p(:r)) «1 p’(:2:) — p"(:c) be a linear transformation. (1) Show that T is nilpotent of index 4. (ii) Find a. polynomial p E PAR) such that F = {p,T(p),T2(p),T3(p)} is a, basis for 733(R). (iii) Calculate the matrix M {1 (T) relative to the basis you find in part (ii). m .3 T3450 and P: (x3,3x3£x,6x»/a,{) 3 .3 a law!) {I [Pa/K) SM-M L“ O 0 O O I“); (T) 1: l O O O 0 1 0 O o O 1 o ...
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