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 DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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; '= (Zrl, ‘3 7‘7." (Li1)\)
. . . 1* .~ Q
Tow; htlecnon vemo. 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 yaxis. 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 oneto—one. T (I, A—& \FF Mfﬂio. 'L \ ‘Z .
[T1 = —z \ 5X & M [T] = O ( Cowms L27. Au PtovovTMUALS
_( q 2' So ‘T' is Dd? AA. 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 realvalued 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. Amirims ; 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 Qiiaé 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= “Wk94:“
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, 11 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+v2 = ”U —V2 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 4tuples (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 ...
View
Full Document
 Spring '11
 PaulaTu

Click to edit the document details