a11 your statements. Remember that you will be graded on what you wnte
Instruction·· Just'fy
· on th e
1
· answer sh eet , NOT on wh at you mtend
.
to write.
Group No:
.1.0
Entry No:
Name:
PtMPI+
KA~TII<
20ZI CS50/2.4
Question-1 (3+1+1 marks):
Consider the homogeneous system of equations AX == O with coefficient from IR, where
A=
-11 -55 1-15 01)
(
2
10 1
8
1 , X ==
1
5
7
0
2
(Xl)
xa
2
X
X
(0)0
and O = 0 .
4
0
X5
(a} By converting A into its row reduced echelon (RRE) matrix find all the free (or independent)
variables in the given system.
(b} Find the rank of A.
(c} Write all the solutions of the given- system AX == ().
0
3
0
0
0
0
0
0
s
s
D
0
2
5'
Li
D
0
-1
-2.
-1
0
0
2.
-1
0
0
"z.
Vi.-
0
0
-'2.
1
'2.
,i
s
(
-1
0
s
. I
0
0
fi3 '-,-.,
5
0
0
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~-'-l,.
0
0
0
0
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0
0
0
0
-y'l,..
0
0
0
0
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e \R
l 7\y ;;- o
v
0
0
I
Instruction: Justify all
·
st
answer sheet, NOT on w'otur a~ements. Remember that you will be graded on what you write on the
a You Intend to write.
Group No: .f.O
Entry No:
Name:
20Z/CS50/2~
Question-2 (2+1 marks):
AAORA
KAR TIK
I
For a~y positive integer n, let Mn(IR) be the vector space of all n x n matrices over the field IR.
Let AAE'Tr(A)
for
Mn(IR).and det(A) denote the transpose ' the trace and the determinant of A, respectively,
(a) Is the subset W1
(b) Is the subset W2
{A E M3 (1R) : Tr(A + 2At) = O} a subspace of Ma(IR)?
{A E M2(JR): det(A) = O} a subspace of M (1R)?
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0
Instruction:
Justify all your st atements. Remember that you will be graded on what you wnte
• on th e
answer
sheet NOT
'
on what you intend to write.
Group No:
Name:
Entry No:
lo
kAATIK
AMM+
I
Question-3 (2 marks):
Let IR.[X] be the vector space of all polynomials over the field IR. Determine whether
X
= {l, 1 + X , l + X + X
3
,
l
+X
th
e subset
3
}
of IR.[X] is linearly dependent. Justify your answer.
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