Mathematics II 2078

Group A 

Attempt any THREE questions.    (10 x 3 = 30)

1. Define system of linear equations. When a system of equations is consistent? Determine if the system 

    -2x₁-3x₂+4x₃ = 5

    x₂-2x₃ = 4

    x₁+3x₂-x₃ = 2

is consistent.            [1+1+8]

2. Define linear transformation with an example.        [1+1+3+5]

    Let , , , 

    and define a transformation T: R²→R² and T(x) = Ax then

    (a) find T(v)

    (b) find x ∈ R² whose image under T is b.

3. Find the LU factorization of

4. Find a least square solution of the inconsistent system Ax = b for

    , 

Group B

Attempt any TEN questions.    (10 x 5 = 50)

5. Determine the column of the matrix A are linearly independent, where

6. When two column vectors in R² are equal? Give an example. Compute u+3v, u-2v where                [1+4]

    , 

7. Let   and define T: R² →R² by T(x) = Ax, find the image under T of

         and 

8. Find the eigen values of

9. Define null space of a matrix A. Let    

         , and 

    Then show that v is in the null A.

10. Verify that 1^k, (-2)^k, 3^k are linearly independent signals.

11. If . Find a formula Aⁿ, where A = PDP^-1 and 

             and 

12. Find a unit vector v of u = (1, -2, 2,3) in the direction of u.

13. Prove that the two vectors u and v are perpendicular to each other if and only if the line through u is perpendicular bisector of the line segment from -u to v.

14. Let an operation * be defined on Q⁺ by a*b = ab/2. Then show that Q⁺ forms a group.

15. Define ring and show that set of real numbers with respect to addition and multiplication operation is a ring.        [2+3]