# Mathematics II 2076

**Tribhuwan University**

**Institute of Science and Technology**

**2076**

**Computer Science and Information Technology ( MTH163 )**

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(e_{2}) = (-5, 2, 0, 0) where e_{1} = (1, 0) and e_{2} = (0, 1);

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

(iii) Is a vertical shear transformation that maps e_{1} into e_{1}-2e_{2} but leaves vector e_{2} unchanged.

Attempt any TEN questions.

Group B (10 x 5 = 50)

5. For what value of h will y be in span {v_{1} , v_{2}, v_{3}} 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 = {b_{1}, b_{2}} and C = (c_{1}, c_{2}) be bases for a vector V, and suppose b_{1} = -c_{1} + 4c_{2} and b_{2} = 5c_{1} - 3c_{2}. Find the change of coordinate matrix for a vector space and find [x]_{c} for x = 5b_{1} + 3b_{2}.

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