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Gujarat Technological University 2010-2nd Sem B.E Computer Engineering ,, Subject code: 110009, Subject Name: Mathematics-II - Question Paper

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GUJARAT TECHNOLOGICAL UNIVERSITY
B.E. Sem-II exam June 2010
Subject code: 110009
Subject ame: Mathematics-II
Date: 23 /06 /2010
Total Marks: 70

Seat No. Enrolment No.

GUJARAT TECHNOLOGICAL UNIVERSITY

B.E. Sem-II Examination June 2010

Subject code: 110009

Subject Name: Mathematics-II Date: 23 /06 /2010 Time: 02.30 pm - 05.30 pm

Total Marks: 70

Instructions:

1. Attempt all questions.

2. Make suitable assumptions wherever necessary.

3. Figures to the right indicate full marks.

Q.1 (a) Attempt any two:

06

i. Solve the following system for x,y and z :

1 3 4 3 2 1 2 1 2

+ + = 30, + - = 9, - + = 10.

x y z

1 0 1 -1 1 1

Find A ¹ using row operations if A =

0 1 0

iii. Find the standard matrices for the reflection operator about the line y = x on R² and the reflection operator about the yz - plane on R³.

^(b) Show that there is no line containing the points (1,1), (3,5),

(-1,6) and ( 7,2).

^(c) i. Find all vectors in R³ of Euclidean norm 1 that are orthogonal

to the vectors u_l =(1,1,1) and u₂ =(1,1,0).

03

02

2 -1 3 4 -2 6

ii. Find the rank of the matrix A =

in terms of

-6 3 -8

determinants. 0 0

0 0

iii. Is

in row-echelon or reduced row-echelon form?

01

04

03

02

05

Q.2

^(a) i. What conditions must b_l, b₂ and b₃ satisfy in order for x_l + 2 x₂ + 3 x₃ = bj, 2 x_l + 5 x₂ + 3 x₃ = b₂, x_l + 8 x₃ = b₃ to be consistent? ii. Is T : R³ R³ defined by T(x,y,z) = (x + 3y,y,z + 2x). linear? Is it one-to-one, onto or both? Justify.

i. Show that the set S = {e^x,xe^x,xV} in C² (-ro, ro) is linearly independent.

ii. Check whether V = R² is a vector space with respect to the operations (u_l,u₂) + (v₁, v₂) = (u₁ + v_l - 2,u₂ + v₂ - 3)

and a(u₁,u₂) = (au_l + 2 a - 2,au₂ - 3a + 3), a e R.

(b)

State only one axiom that fails to hold for each of the following sets W to be subspaces of the respective real vector space V with the standard operations:

[D] W = {A_nxn | Ax = 0x = 0},

[E] W = {f|f(x)< 0, Vx},

Check whether S = {sin(x + 1),sinx,cosx} in C(0,ro) is linearly independent.

[A] W = {(X,y) | x² = y²},

[B] W = {(x,y) | xy > 0},

[C ] W = {( x, y, z) | x^! + y² + z^! = 1},

V = R2

V = R3

V = M

n x n

V = F (ro, ro) . ro'

02

ii.

Q.3

(a)

03

i. Determine whether the following polynomials span P₂ :

p₁ = 1 x + 2x², p₂ = 5 x + 4 x², p₃ = 2 2 x + 2 x².

ii. Show that S = {1 t t³, 2 + 31 +1² + 21³,1 +1² + 51³} is

linearly independent in P₃.

i. Find a standard basis vector that can be added to the set S = {(1,2,3), (1, 2, 2 )} to produce a basis of R³.

ii. Determine whether b is in the column space of A, and if so, express b as a linear combination of the column vectors of A if

(b)

03

	" 1 1 1"		" 2 "
A =	1 1 1	and b =	0
	1 1 1		0

(c)

i.

If A is an m x n matrix, what is the largest possible value for its rank.

Find the number of parameters in the general solution of Ax = 0if A is a 5 x 7matrix of rank 3.

01

ii.

OR

Q.3

(a) i. Find basis and dimension of

03

W = {(a₁,a₂,a₃,a₄) e R⁴1 a₁ + a₂ = 0,a₂ + a₃ = 0,a₃ + a₄ = 0}.

ii. Find a basis for the subspace of P₂ spanned by the vectors

1 + x, x , 2 + 2 x , 3 x.

^(b) i. Reduce S = {(1,0,0 ), ( 0,1, 1), ( 0,4, 3), ( 0,2,0 )} to obtain a 03

03

basis of R .

ii Find a basis for the row space of A and column space of A if ₀₃

A =2130

13 2 0 matrices.

1 2

(c)

02

Show that S =

is a basis

1 2

1

⁰

₁

1

O

²

1

⁰

1

3 1

00 1 2

. Also verify the dimension theorem for

^(a) i. Compute d ( f, g) for f = cos2nx and g = sin2nx in

1

V = C [0,1] with inner product (f, g) = J f (x) g(x) dx.

0

ii. Find a basis for the orthogonal complement of the subspace of R³ spanned by the vectors v₁ =(1, -1,3), v₂ =( 5, - 4, - 4) and v, =( ⁷, - ^6,2).

03

(b)

^, (^0,1,0)}

03

If 4,0,

5 5 .

1,0)}. Express w = (1,2,3) i

i. Let W = span

in

the form of w = w₁ + w₂, where w₁ e W and w₂ e W . ii. Define algebraic and geometric multiplicity. Show that 1 0 0'

03

A =

is not diagonalizable.

1 2 0 -3 5 2

^(c) Show that P₃ and M₂₂ are isomorphic.

OR

03

Q. 4

(a)

03

Let R³ have the Euclidean inner product. Transform the basis S = {(1,0,0), (3,7, - 2), (0,4,1)} into an orthonormal basis using the Gram-Schmidt process. .

i.

ii. For U =

u₁ u₂

v v

12

and V =

l^u3 ^u4 J

1

^v

1

in M₂₂, define

{U, V) = u₁v₁ + u₂v₂ + u₃v₃ + u₄v₄. For the matrices A and B, verify Cauchy-Schwarz inequality and find the cosine of the

02

" 2 6"		"3 2"
	, B =
_L1 -3_		1 0 _

angle between them, if A =

(b) i. Find the least squares solution of the linear system Ax = b and

find the orthogonal pro			ection of b onto the column space of
	"1	¹"		" 7"
A where A =	-1	1 ^__	and b =	0
	-1	2		-7

ii. Find the transition matrix from basisB = {(1,0),(0,1)} of R² to basis B^f = {(1,1), ( 2,1)} of R².

03

(c)

03

1 + i 1 + i

2 2 1 - i -1 + i

For the matrix A =

show that the row vectors form

L 2 2 J an orthonormal set in C². Also, find A"¹.

(a)

04

i. For the basis S = {v_l,v₂,v₃} ofR³, where v_l =(l,l,l),

v₂ =(l,1,0) and v₃ =(l,0,0), let T : R³ R³ be a linear transformation such that T(v_l ) = (2, -1,4), T(v₂ ) = (3,0,l),

T(v₃) = (-1,5,1). Find a formula for T(x_l,x₂,x₃) and use it to find T ( 2,4, -1).

ii. Let T :M₂₂ R and T₂ :M₂₂ M₂₂ be the linear transformations given by T_l(A) = tr(A) and T₂ (A) = A^T.

a b

02

Find (T ⁰ T₂)(A) where A =

c d

(b)

04

1

-1

Find a matrix P that diagonalizes A =

and hence find

A¹⁰. Also, find the eigenvalues of A².

Let T : R⁴ R³ be the linear transformation given by

(c)

04

) = (w_l,w₂,w₃) where w_l = 4x_l + x₂ -2x₃ -3x₄,

^T ( x

^{x , X}2 ^{, X}3 ^{, x}4,

w₃ = 6 x_l - 9 x₃ + 9 x₄. Find bases for the

w₂ = 2x_l + x₂ + x₃ - 4x₄, range and kernel of T.

OR

Q.5

^(a) Let T : R² R³ be the linear transformation defined by T ( x_j, x₂) = ( x2, 5 xj +13 x2, 7 xj +16 x2).

Find the matrix for the transformation T with respect to the bases B = {(3,l)^T , (5,2)^T } for R² and

04

B = {(l, 0, - l)^T , (-1,2,2)^T , (0,1,2)^T } for R³

(b)

04

i. Let T : R² R²be defined by T(x,y) = (x + y,x - y). Is T one-one? If so, find formula for T^-l (x, y ).

% ⁵⁷⁶

-420

0

01

ii. Find eigenvalues of A =

. Is A

0

0.6

J3

(c)

05

invertible?

Translate and rotate the coordinate axes, if necessary, to put the conic 9 x² - 4 xy + 6 y² -10 x - 20 y = 5in standard position. Find the equation of the conic in the final coordinate system.

4

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