Another F2009 Sample Midterm 2 test with solution

# Another F2009 Sample Midterm 2 test with solution - Linear...

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Unformatted text preview: Linear Algebra F09 Name: Page 1 of 7 (1) (5 marks) Find parametric equations for the line passing the point P (1, 1, —1) and parallel to the two planes given below: 233—y+z=0, a:—2y+z=3 VT; '= (Zr-l, ‘3 7‘7." (Li-1)\) . . . 1* -.~ Q Tow; htlecnon ve-mo. FOL uue 1 -_ Ext/i, = L J -_- ( \ ,_\ ,3) ‘L —\ \ t '2 \ Mow (mm—2.3 = (\,\,—\‘)+k( \,—\,—7a) (2) (4 marks) Find the steady state vector q for the following regular tran- sition matrix: P _ 1/3 1/5 _ 2/3 4/5 \A’AW 1 50¢“ THAT Pi=i c=-> (1“?) is 9 1| 3,1. 1—9: [2’3 ”‘1 com ii ﬁoV 4/3 V'S' ”was 0 o 0 LGT 962‘s: GET \$5: 130—5 . u, _ \ = E. 3" Hills/ls 7: "a so l'iiybl Linear Algebra F09 Name: Page 2 of 7 (3) (4 marks) Say T : R2 —> R2 is the linear operator deﬁned by: a reﬂection about the line y = as followed by a clockwise rotation by an angle of 3% followed by an orthogonal projection onto the y-axis. Use [T] = [T(e1) | T(e2)] to get the standard matrix of T. (NOTE: for full marks, you must use this theorem.) (4) (3 marks) Let T : R3 —> R3 be the linear operator deﬁned by T(a:1, 3:2, :63) = (2331 + 332 + 2333, —2331 — 332 + 5333, 4331 + 2:132 + 7333) Determine if T is one-to—one. T (I, A—& \FF Mfﬂio. 'L \ ‘Z . [T1 = —z -\ 5X & M [T] = O ( Cowms L27. Au Ptovov-TMUALS _( q 2' So ‘T' is Dd? A-A. Linear Algebra F09 Name: Page 3 of 7 (5) (5 marks) Determine Whether the set S = {M1, M2, M3, M4} is a basis for the vector space V of 2 X 2 WW matrices, Where 1—1 1—1 1 0 —1 0 M1=i1oiM2=i2oiM3=iloiM4=io1i S 1'; A Ms'xs Fez \i {FF S 15 u‘mmn. 1339 S stusU. caecu'uc. S is Lin. fuss. LET \L‘H\+‘4LM7_+\L3M3+quR = O , GET SYSTEM \4\ +‘4L "4‘3 ’k“ :0 _.k_\ ”kl =0 k\ +Zkz+k3 = 0 Li -0 Eri'uel. Use GAOSSiAu gunman-(09: 2‘ L. \c, lea l\ l\ ‘o —0‘ 29w :3 i ‘0 T : Ger kph—Jain“) is rue 03"” - . l ‘ ’L \ 0 £6046 o o [ ~l o SoLUﬂoIJ o o o \ ° ° ° ‘ 0 5° S (5 Lil). Mo. 0L 05g bET. cf— mewidem mmu'x , CALL W A- l ! i 0 9.39.3 MA‘; —\-—\oO : :41“) l 1. \ O '” o o a \ MAio c) \$761114 H45 mow “me nwi. SoUu. So 3 is mm“). Linear Algebra F09 Name: Page 4 of 7 (6) (6 marks) Let V = F(_OO7OO) be the vector space of all real-valued functions with the addition deﬁned by ( f + g) (as) = f (a:)+ 9(33) and scalar multiplication deﬁned by (kf)(a:) = k(f(a:)), and let W be the set of all functions f in F(_OO7OO) such that f(a:) 2 a12‘” + a22_"" + a3 Where a1, a2, a3 6 R. (i) Show that W is a subspace of V. Cue‘cv. masons opium. Ami-rims ; u: if \U Tue“ 6+3 6“}. So SAY .2\qanJ. TUAT 1% gm): 4‘ 2X+erx +0.1 & 36c) : ‘0‘ 15‘ -K- bzz'y —\- b3 helm Qg‘s Sass 6(2 mu qux) = ﬂaw) = (who? [email protected];+(o132"‘+ (“3“le ‘ , g...— , W em. em 6'1 S9 Qii-aé VJ Cuecv. cxosubé OWNER. \$04441. MULT‘AJ: 1F £60.) & hellﬁ’ueﬂ lLQéhl 39 5A7 (2 As Awe 8< law. @23ch WWanggu’ugg) so iggw W’ (L €10. él em 30 \J is A \$0\$SOACE 0C C("Nﬁ‘o (ii) Give a basis of W and state the dimension of W A (5956 is {1”, 2" , 413 s a who) =3. Linear Algebra F09 Name: Page 5 of 7 (7) (8 marks) Consider the following matrix A and its 7“.7“.e.f., call it B 261170 @3002 0 A_ 1 3 3 0—1 1 R_ 0 0®0—1 1/3 41271527‘ [email protected]—1/3 13—1—1—10J [00000 0 (a) Give a basis for the rowspace of AT w(Av)=CaC¢«mCA), \$0 A (\$166 is [LE1— \ H 1"5‘['\ 3 ‘7 —-\"})[_\ o L “jg (b) Give a basis for the nullspace of A “ed )(7’=\$J Kr=k) X5=A I GET Xq= “Wk-94:“ X‘s: ﬁ-‘Jéu. X\‘ “33—215 -3, —7_ o _ I a a so 3" .— 0 S 'l' l l; 'l "/3 w 0 ”V V; o l o o o I —‘> -7. 0 So A ems m MﬂmuM) \s 3, ﬁ’ 3,} 3 "H ’ V3 0 O ‘2 (0) State the following: (i) TCLTLk<A> = ‘3 (ii) nullity(A) = 3; (111) Tank(AT) 3 (iv) nullity(AT) = l Linear Algebra F09 Name: Page 6 of 7 (8) (5 marks) True / False. Indicate Whether the following statements are always True or sometimes False. (a) Elementary row operations do not change the column space of a matrix True E/False (b) If AX = 0 has only the trivial solution and A is n X n, then nullity(A) = 0 3/ True False (c) If T(u) = 0 and T is a linear, 1-1 transformation, then u = 0 True False (d) The planes 2a: — y + 3,2 — 2 = 0 and 3a: — 22 + 1 = 0 are perpendicular to each other if True False (e) With u,V E R”, then ||u+v||2 = ”U —V||2 ifu i V 3’ True False (f) If A is an n X m matrix, and nullity(AT) = 2, then TCLTLk<A> = m — 2 True 3/ False (g) The set of all 4-tuples (a, b, c, d) With positive entries is a subspace of R4 True 3/ False (h) If S is a subset of vectors in a vector space V, and V E V is not in span(S), then span(S) = span(S U {V}) True j False (i) If W is the space of n X n diagonal matrices, then dim(W) = n2 True 3/ False (j) If {V1, V2,V3} is a linearly dependent set, then so is {V1,V2} True Bf False ...
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