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 DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: :DT' A‘jarml, SOLUTmNs Fog ExAm~l Linen A‘jelom—}
low631 51””‘320‘0' 1. State if the following are TRUE or FALSE. Give a brief reason for your answer. (a) If a system [Alb]
solutions. False. ‘l‘eie 5:3ng can 52 inmziskmf , for {M (D2 3 H 3 w“;
[ @333 ‘elﬁd I;{\m’t(grim)0“;t m. has at least one free variable then it will have inﬁnitely many (b) If a system [Alb] has a pivot in each row then the system has a unique solution. False. Hm $851an an rue/Hoe. «New (M9. errand)
' m' ”(Lib FWQK
W5 5;; mfx +18 ”"4143, N1 “*1" [(9 2 3 1"] 9M. w & own” 1 2 3
(c) The rank of the matrix A = I: 2 4 6 J is 2. Fake, 1 1 2 ——1
(d) The two lines given by r = (0) +15 (1) and r = (0) + S ( 1 ) are orthogonal.
1 1 2 0 Time. 1 19
(e) Let v = [0] and u = [0] . The linear span of {V, u} is a line. Tr e, 0 0 latcﬁ‘z. ! u " "'   a
m . . : Clu+czv '" 0 ‘ W33 W” W ° / ° [Wm hatn2, MA dmﬁm saw. m (a). will be an inconsistent 0 0 2
(f) Let A = (0 l 7) . There exists a vector b such that [Alb]
l 0 1
system. 0 I 7 SM? 0 o 2,
W
eckeLon 12”, 31M“
7% raMkCM=3 O pl'vo‘f
4—dimensional solution. 0 ‘
2
hue " R'sz— O
"==a==‘: ——9 O O O C 0
O O 0 0 0
r 4r 1 r 2. Suppose u, v E R" such that ”u” = 3, Hu— 3v” = 2 and “v“ = 1. Find the angle between
the vectors u and v. . we, mat ro Illavllz: (a'3V)°(“"3") L
r. “unz— 5&1:qule 4 396(aV)+‘l 3. Let A the coefﬁcient matrix and the constant (right side) vector b be given by (a) Using Gaussian elimination on the augmented matrix [
solution in vector form. 423442‘1 l 2. 3 3 _R2+123 ‘2 3 3 ~R.+R3 $1 5 '5 Alb]. Find the complete 2L,74..______§247‘1.____.,00®'2
36w 7 '2'3 3 “290°“: °"°°
0000 0000 0000 —2.
x 3—23“. 9
z _2_ «2. o (b) Show that W = Not. [33] z ml? 9» wa‘lm M3” 3 70
7D =3 We SFQAA%CO€I'53.A' 4. Let A and b be as below. Find the row echelon form for the augmented matrix to answer
the following questions. 1 0 0 4 2
1 1 0 0 0
A‘ 1 0 1 0 b‘ 0
1 2 3 m k
For What values of m and [C will the system [Alb] have
(1). no solution
(2). inﬁnitely many solutions.
(3). a unique solution.
0 O 2.
‘RN'RJ I o OH 1 "ZRZ+RH I H
A_______«, OloHL________,OIOHz
"RW? o a 142. 00 l "4?—
_RI* ‘1 O 2_ 3mHlR—L a o 3M9 th.
I O 0 L4 2.
.3123HZ‘1 o I o —q 2_
~———""'"’ 0 o I '¢ ~z
o o gnu16 k+8 '1) 7F" ““0 SOM'W’ m+lb= 0 Mel k+57€°
:5 m:—'lé M k9é’2’ u m 706’
. I l m ”(Hal . OA’ XMU‘M—(F'a'ww A P
11) {01W ( (MM are.) M+K9=O M k+5=° 5. For the following, support your answer with a very brief reason. Without reason NO
CREDIT will be given. (a) What will be the reduced row echelon form of the 5 X 3 “coefﬁcient” matrix A7 if
the homogeneous system [AID] has only the zero solution. ' Atssxaam/Asa .
ﬁts/Newt. Salad? [Al63 ﬁvoﬁumm 2)?“ 3' OO O I
O
a l
Tre§ 05A = E
O 000.... (b) Suppose R is a 6 x 5 “reduced” row echelon matrix with rank 4. Give a description
of a constant vector b for which [ Rlb] will always be an inconsistent system. Tank'sLl :9 onlj Ll PWO‘B :2) 011161413. W. noooat L.
Z'O‘H‘ b2 vfmimmsts ,
Ogleu [,3 b E
01): b O.
000 w or T79
000%% 3‘ 5 C
5 (0) Suppose A is a 3 X 3 matrix and b is a vector in R3 such that [A]b] is inconsistent.
Does there exist a vector c in R3 such that [A]c] has a unique solution? l? [AIL] 1's theoncileMf' =§ Wmfs apn'vo't’t'n ladMum. «>thqu
\ ._. ' I 3. S ‘b"
Simon [A] b] Is a 3X“! Mai/11x “—3“ JUNE 5 3 i ' ' Us".
As new ohm) tum. L6 aha{j ax Fwd m (ti/(041’
ﬁl’llm =e «”ka ~32. 6. Let @100
\Imcop—t 6 Suppose A is a matrix such that [Alu] and [Alv] are both IN CONSISTENT systems Is
it possible that [Al(u + v)] is a consistent system with inﬁnitely many solutions If “YES”, then give an example of such a matrix
conditions given. If “NO” A and show that it satisﬁes all the , then give a reason why would such a matrix A not exist. l S .._._.3 O O O
a q o o o
(5" (S' (5' O O 0 7. The complete solution of a system of linear equations with augmented matrix [Alb]
given by X‘ l 1 2
X2 csolution = 0 +0 1 + d 0
X3 —1 0 1 is Since, um. Wm $613., is in 3D spam.
=9 has. " ta “nigh, ﬁuvw'adok’s.
2% T: nz..1 . M.
MC; mmmcﬁm an “M , exccf‘t' M 7'“? we Can cansmd: 0L W simpﬂ twin: ...
View
Full Document
 Spring '10
 Dr.AnuragAgarwal
 Linear Algebra, Algebra

Click to edit the document details