USN:

M. S. RAMAIAH INSTITUTE OF TECHNOLOGY(Autonomous Institute , Affiliated to VTU)

Semester End Examinations - January 2009

Course : M.Tech (Digital Electronics & Communication)Subject Code: MDLC11 Subject: Advanced Engineering Mathematics

Maximum Marks: 100 Duration: 3 Hours

Instructions to the Candidates: Answer any 5 full questions

1. a) The probabilities of 0, 1, 2 , 3 power failures in a certain city during the month (4)of July are 0 . 4, 0.3, 0 . 2 and 0.1. Find the mean and variance of this probabilitydistribution.

b) The auto correlation sequence of a random phase sinusoid is (6)r,(k)= ' A2 cos(kwo). Find (i) 2 x 2 auto correlation matrix R,,(ii) Eigen values of R., (iii) Determinant of R.

c) The power spectrum PY(etw) of a WSS process is a real valued, positive and (10)

periodic function of co. Show that P,,(z) can be factored into the form

Px(z) = o Q(z)Q*(1/z*)

2. a) Determine the solution set of the given system in parametric vector form (5)x, +3x2-5X3=4

x,+4x2-8x3=7-3x, - 7x2 + 9x3 = 6

b) Prove that any set { v, .......vp } in R" is linearly dependent if p > n. (5)

c) Let T:R3-R3 be the transformation that reflects each vector x=(x,, x2, x3) (5)through the plane x3 = 0 onto T (x) = (x,, x2, -x3). Show that T is a lineartransformation.

d) A Givens rotation is a linear transformation from R" to R" used to create a zero (5)entry in a vector. The standard matrix of Givens rotation in R2 has the form

n -b 2 , 4 5

b a, a + b = 1. Find a and b such that 3 is rotated into 0

[] .

3. Given

A=

4 -1 -1

-1 4 0 -1

-1 0 4 -1 -1

1 -1 4 0 -1

-1 0 4 -1 -1

1 -1 4 0 -1

-1 0 4 -1

-1 -1 4

b=

5150100102030

(20)

The missing entries in A are zero. (i) Construct LU factorization of A and

compute LU-A to check your work. (ii) Use LU factorization to solve Ax=b.(iii) Compute A' .

4. a) Given v, and v2 in a vector space V, let H = span (VI, V2). Show that H is asubspace of V.

(5)

b) Determine whether w is in the column space of A, or the null space of A or both (5)where

w=1

-3

A-

7 6 -4 1

-5 -1 0 2

9 -11 7 -3

19 -9 7 1

c) Find (i) bases for col A, Row A, Null A (ii) dim Null A, dim Row A, rank A, (10)rank A t for the given matrix A.

I1 1 -3 7 9 -911 2 -4 10 13 -12

A= 1 -1 -1 1 1 -31 -3 1 -5 -7 3

1 -2 0 0 -5 -4

5 (a) Solve the initial value problem using eigen values and eigen vectors (10)dv/dt = 4v -5w v=8 at t=0dw/dt = 2v -3w w=5 at t=0

b) The Fibonacci sequence starts with Fo=1, F,=3, then Fk+2 = Fk+( + Fk. Find the (10)new initial vector Uo , the new coefficients c=S-'U0 and the new Fibonaccinumbers. Show that the ratios Fk+,/Fk will approach golden mean.

6. a) Apply row reduction operations to matrix A to produce an upper triangular (8)matrix U and compute the determinant.

1 2 -2 02 3 -4 1

A= -1 -2 0 20 2 5 3

b) Show that the determinant of A is zero for any values of a, b, c (2)1 1 1a b c

A=b+c c+a a+b

c) Find matrices S and A such that A has a factorization of the form A=SAS-' (10)

where S is invertible and A is diagonal. Find the matrix B such that B3=A.2 2 1

A= 1 3 1

1 2 2

7. a) Find M that achieves M-'BM = J for the matrix B where J is Jordan form. (6)

0 0 1B= 0 0 0

0 0 0

b) Apply the Gram-Schmidt algorithm to the columns of the matrix A. (i) WriteA=QR where Q is a matrix with orthonormal columns, R is upper triangular.(ii) Compute the matrix of the projection onto the column space of A. What is

the distance of the point (1, 1, 1, 0) to this column space?

A=

(10)

c) Determine whether the matrix A is positive definite, negative definite, semi- (4)

definite or indefinite.

1 2 3A= 2 5 4

3 4 9

8. a) Find a singular value decomposition of the matrix A. (10)

1 -1A= -2 2

2 -2

b) Solve Ax=b by least sqaures. Find P = Ai if (10)

1

A= 0

1

1

1

b= I

0

Verify that the error b-P is perpendicular to the columns of A.

3