Mathematics II 2076

Group A (3 x 10 = 30)

Attempt any THREE questions.

1. When a system of linear equation is consistent and inconsistent? Give an example for each. Test the consistency and solve the system of equations: x - 2y = 5, -x + y + 5z = 2, y + z = 0

2. What is the condition of a matrix to have an inverse? Find the inverse of the matrix If it exists.

3. Find the least-square solution of Ax=b for 

4. Let T is a linear transformation. Find the standard matrix of T such that

(i) by T(e1) = (3, 1, 3, 1) and T(e₂) = (-5, 2, 0, 0) where e₁ = (1, 0) and e₂ = (0, 1); 

(ii) rotates point as the origin through radians counter clockwise.

(iii) Is a vertical shear transformation that maps e₁ into e₁-2e₂ but leaves vector e₂ unchanged.

Attempt any TEN questions.

Group B (10 x 5 = 50)

5. For what value of h will y be in span {v₁ , v₂, v₃} if 

6. Let us define a linear transformation. Find the image under   

7. Let Determine the value (s) of k if any will make AB = BA.

8. Define determinant. Compute the determinant without expanding 

9. Define null space . Find the basis for the null space of the matrix 

10. Let B = {b₁, b₂} and C = (c₁, c₂) be bases for a vector  V, and suppose b₁ = -c₁ + 4c₂ and b₂ = 5c₁ - 3c₂. Find the change of coordinate matrix for a vector space and find [x]_c for x = 5b₁ + 3b₂.

11. Find the eigen values of the matrix 

12. Find the QR factorization of the matrix 

13. Define binary operation. Determine whether the binary operation * is associative or commutative or both where * is defined on Q by letting . 

14. Show that the ring is an integral domain.

15. Find the vector x determined by the coordinate vector where