V3solutions

V3solutions - Linear Algebra F09 Name Page 1 of 7(1(5 marks...

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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 .a J ““(1‘-\,\) A,=(\,—2,l) ' ' L F ME '3 ' Ax" -= L k 1mm memo» uecno ozu — n. n1, 1 _\ | = 0,4,5) \ —7. \ (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 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 377T 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 Cs A~4 2 l L W: -2 -. s :0 cam Li 1 7 so TR 1337 L—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 matrices, Where 1—1 1—1 1 0 —1 0 Ml-il oil‘s-[2 oil‘s-i1 aim-i0 1i g (s A @616 s is Lin. inset. 2. (5m- suiicc ALL...U= Li owl we “Ave Lkueuozs in S, (T is emcee: To SHM S (5 LG). fume .5 l ‘ So cuéct/LS \S ummw’. LET \<\M\‘L\41M14k3ms+kfiflch L‘+kL-\—k3 "'kq :0 ‘k‘—k1 = O \L\*1kl+k3 = O O 2. svsTtM HA5 ow! Tet “w.le Sm‘u 1% M kfo ‘ ‘ CgCrcl = 4750 So Maia; ‘43=i¢«=0" UULQUE . 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) = f (a:)+ 9(33) and scalar multiplication defined by = k(f a: t W be the set of all functions f in F(_OO7OO) such that f(a:) = ale";+a2e_‘” here a1, a2 6 R. [67" is the exponential function] (i) Show that W is a subspace of V. @ CHECK cuss»: gum Amine») \F £$oae u) Tum 5m Qatar}. “Eqékd =2 {och Q\€x-\-Q,_e"‘ 86d = lake‘u- (016” MC QulDUQI‘laLé (L ed. élL (ii) Give a basis of W and state the dimension of W A M515 is i exie'x 24 so 4413031- L k'acz"':o) Q$=A Q|=AIQ>=O £ny gamut. + a. 00x Linear Algebra F09 Name: Page 5 of 7 (7) (8 marks) Consider the following matrix A an its 7“.7“.e.f., call it B X L \(3, liq V: Y; [ 6 1 1 7 0 Q 3 3 0 1 1 0 14— 12 7 1 5 2 7 R_ 0 l 3 — —1 OJ 1 0 Ir ( a) Give a basis for the rowspace of AT r WA 1C7- ‘ H \"S,Y_\ "a '1 —\'\J[L-o 1—4—33 (b) Give a basis for the nullspace of A LET x1=9:') XIV-'32 x£=m 95A x“: -(-i$'\"éU~ x5: 5 —Jéu. x\= ‘15 X\ “.3 —Z O ‘41 A 0 0 so A 951$ m mva‘l "5 :5 : 0 2k + \ 5+ '33 on -3 A g " O 4* V3 0‘ T -‘ 15 o \ o o ‘q ) l 1; o o l g ) l 03 O (C) State the following: (i) TCLTLk<A> = 3 (ii) nullity(A) = E (iii) rank(AT) = 3 (iv) nullity(AT) = l (W MCEl+Afi£ECBF #4 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 3’ False (b) If AX = 0 has only the trivial solution and A is n X n, then nullity(A) = 0 8/ 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 {True False (e) With u,V E R”, then ||u+v||2 2 Mn —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 Z/False (g) The set of all 4-tuples (a, b, c, d) With positive entries is a subspace of R4 True 31/ 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 3/ True False (i) If W is the space of n X n diagonal matrices, then dim(W) = 712 True fl False (j) If {V1, V2,V3} is a linearly dependent set, then so is {V1,V2} True B/False Linear Algebra F09 Name: Page 7 of 7 BONUS (5 marks) Let T be a 1-to-1 linear operator on R3. Show that if S = {V1,V2,V3} is a basis of R3 then T(S) = {T(V1)7T(V2)7T(V3)} is also a basis of R3. ...
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V3solutions - Linear Algebra F09 Name Page 1 of 7(1(5 marks...

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