question paper contains 8 printed pages.
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[This
(23) Your Roll INoduLZ.2023
Question Paper : 1271 F
Sr. No. of
Unique Paper Code : 2352571201
: ELEMENTARY LINEAR
Name of the Paper
ALGEBRA
Name of the Course : B.Sc. (Prog.) DSC-B2
Semester BARDNy
Duration:3 Hours Maximum aks : 90
LIBRARY
Instructions for Candidates De
i
1. Write your Roll No. on the top immediatelyon receipt
of this question paper.
from each
2. Attempt all question by selecting two parts
question.
3. All qquestions carry equal marks.
4
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2
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vectors in R then prove that:
1. (a) If x and y are
l|x + yll s l|xl| + Iyll
norrn ofa vector. Find a unit vector in the
(b) Define
1 1 2|
vector
same direction as the
vector longer or
Is the normalized (resulting)
shorter than the original? Why?
Gaussi.an elimination method to solve
(c) Use
Give the
föllowing systenms of linear equations.
solution set 1s
complete solution set, and if the
infinite, specify two panicular solutions.
3x + 6y -9z = 15
2x+4y - 62 10
--2x 3y +4z = -6
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3
-arI|BRARY
2. (a) Determine whether the 3
tWo
equivalent? matrices are roW
3 -1 21
2 10
L5 -1 5! 2 0 4!
(b) Find. the rank of the
following matrix.
-1 0
0 0 2 3
4 0 -2 1
3 -1 4
(c) Express the vector X = [2, -5, 3] as a
linear
combination of the vectors a,= [l, -3,2], a, |2;
-4,-1], and a,= [1, -5,7] if possible.
P.T.0.
(a)
weIIIIC UIIU
TOIIOWII8 TeriIIs (aily 10WI). (+)
() Metamorphosis
Altempt any five questiouS
characteristie po
poy
Determine the
(a)
following matrix.
2]
3 w
1
-2
of the form (a, b, 0,
that the set of vectors
(b) Show
forms a subspace of RS
in R
C, a - 2b + c]
under the usual operations.
-5x, x-r},
use
2.2, I -- 4.x2, 12
(c) For S= + a simplified
Method to find
the Simplified Span
the vectors in span(S), where
general form for all
the set of all
S is the given subset of P, ,
equal to 3 with
than
polynomials of degree less or
real coefficients.
4. (a) Use the Independence Test Method to determine
independent or
whether the given set Sis linearly
linearly dependent.
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s = {1, -1, 0, 2], [0, -2, 1, 0], [2, 0,
the -1, 1}
() Let the subspace
W of R° be the solution set to
the matrix equation AX = 0 where A is
2 11 0 -11
b, 0, 2 -1 1 3
1 -3 -1 1
2 9 4 -1 -7
use
Find the basis and the dimension for W, Show
lified
that dim(W) + Rank(A) = 5.
where
bf all
(c) Show that P. n
the set of all polynomials of degree
Bwith
coefficients, is a
less than or equal to n with real
operations of addition
vector space under the usual
and scalar multiplication.
ent or
S. (a) f:R ’ R given by
Consider the mapping
P.T.0.
petween uIt piavs
I5 tneII InSereu
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f(la,, az, a ) = [4,, az -ag]
Prove that f is a linear transformation.
(b) Find the matrix for the linear transformation
L:P ’ R given by
L(azx³ + ayr + ajt a) = [ag t ay, 2ay, az - ao]
With respect to the bases B = (*', r, x, 1) for P,
and C = (e,, e,, e,) for R3
(c) Consider the linear operator L: R" ’ R" given by
L(la,, az, *", an) = [a,, az, O.,0]
pacitor. of
P.T.0.
Show
one-to-one is that
L
onto and
operator.
-sin1|$ -|(C)
Lsin
fcos9
linear Consider
the (b)
given
by R² ’operator
R2L:
for basis the Find
L.kernel
of
J|X3. 15 -3 3 by
-13||X2 3 -2
5 1
transformation linear Consider
the (a) 6.
given R ’ RL:
L.range
of and Lkernel
of the Find
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7
isomorphism? an LIs
P3
pEwhere L(p)=p' by
ven transformation
P2 ’L:P3 linear Consider
the (c)
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