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USN

MDLCII1 M S

M S RAMAIAH INSTITUTE OF TECHNOLOGY(AUTONOMOUS INSTITUTE, AFFILIATED TO VTU)

BANGALORE - 560 054

SUPPLEMENTARY SEMESTER EXAMINATIONS - AUGUST 2012Course & Branch

Subject

Subject Code

M.Tech. - Digital Electronics andCommunicationAdvanced Mathematics

Semester : I

Max. Marks : 100

: MDLC11 Duration : 3 Hrs

Instructions to the Candidates:• Answer one full questions taking at least one from each unit.

c)

Solve the following linear system of equations. (07)

x1 -3x3 =8; 2x, +2x2 +9x3 = 7; x2 +5x3 = -2

For what values of h will Y be in span V1 ,V21 V, I if (07)

1 5 -3 -4

v1= -1 ; v2 -4 ; v3= 1 and Y= 3

-2 -7 0 h

0 1 2

Find the inverse of the matrix A= 1 0 3 if it exists. (06)

4 -3 8

Determine the existence and uniqueness of the solutions to the system. (07)

3x2 - 6x3 + 6x4 + 4x5 = -5

3x1- 7x2 + 8x3 - 5x4 + 8x5 = 9

3x1- 9x2 +12x3 - 9x4 + 6x5 =15

Determine if the columns of the matrix form a linearly independent set. (07)

Justify each answer.

c) Find the standard matrix A for the dilation transformation T (x) = 3x, for x (06)

in R2.

UNIT - I

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UNIT - II

Given v, and v2 in a vector space V, let H=Span{v1,v2} show that H is a (10)

subspace of v.

b) For the matrix below ( i) find rank A and dim Null A ii) Find bases for col A (10)

and Row A

A=

2 -1 1 -6 8

1 -2 -4 3 -2

-7 8 10 3 -10

4 -5 -7 0 4

5b+2c

4 a) Let W be the set of all vectors of the form b

c

where b and c are (10)

arbitrary. Find the vectors u and v such that W =Span{u,v} . Show that W

is a sub space of R3.

b) Let b,13 ' b2

[4 ]

9C2

bases for R2 given by /3 = {b1,b2} and C = {c1,c2} .

Find:

i) Change of Co-ordinates matrix from C to f3 and

ii) Change of co-ordinates matrix from 8 to C.

and consider the (10)

UNIT - III

a) Find all the eigen values and corresponding eigen vectors for the matrix (10)

4 0 1

A= -2 1 0

-2 0 1

b) Let A=7 2

-4 1. Find a formula for Ak given that A = PDP-` find A5 . (10)

a) Diagonalize the matrix, if possible and verify that (10)

A = PDP-' where A =

1 3 3

-3 -5 -3

3 3 1

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b) The mapping T: P2 --> P2 defined by T(ao+a,t + a2t2 )= a,+2a2t then (10)

i) show that T is linear transformation

ii) Find the B - matrix for T, where B is the basis {l , t,t2} .

iii) Verify that [T (p)],, _ [T]fl [P]fl for each Pin P2.

-2

U2 = 17. a) Let ul = 5

11 L3J

closest point in W to Y and also find its distance.

r21

I] rol [01

b) Let X, _1

1

1

x2 =1

1

1]

Unit - IV

1Y= 2 and w=Span {u,,u2} then find the (10)

x3 =0

1

1

. Then {x,,x2,x3} is clearly (10)

linearly independent and is a basis for a subspace w of

orthogonal basis for W.

8. a) Let u, _ 5

-1

U2 = and y =

R4. Construct an

'-1

2 . If {u,, u2 } are orthogonal (10)

-3

basis of w=span {u,,u2} . Then write Y as the sum of a vectors in w and

vector orthogonal to w.

b) Find a QR factorization of A =

Unit-V

9. a) Orthogonally diagonalize the matrix A =

equation is O = -23 + 1222 -21/1-98

3 -2 4

-2 6 2

4 2 3

(10)

whose characteristic (15)

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b) For x in R3, let Q(x) = 5x, +3x2 +2x3 -x1x2 +8xzx3 . Write this in quadratic (05)

form as XT AX .

1 -1

a) Find a singular value decomposition of A= -2 2

L 2 -2

(15)

b) LetQ(x)=x; -8x1x2-5x2 . Then compute the values of Q(x) for (05)2

-31 [ 2 ]

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