Mathematics II Model Question

Group A

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

1. What is pivot position? Apply elementary row operation to transform the following matrix first into echelon form and then into reduced echelon form:

2. Define linear transformation with an example. Check the following transformation is linear or not?  be defined by T(x, y) = (x, 2y).Also, let T( x,y)= ( 3x+y, 5x+7y, x+3y). Show that T is a one- to-one linear transformation. Does T maps onto ?

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. Compute u+ v, u-2v and 2u+v where .

6. Let , and define  be defined by

T( x) = Ax, find the image under T of  and .

7. Let and . What value(s) of k, if any, will make AB = BA?

8. Compute det A, where .

9. Let H be the set of all vectors of the form . Show that H is a subspace of .

10. Find basis and the dimension of the subspace

    .

11. Find the eigenvalues and eigenvectors of .

12. Define orthogonal set. Show that {u₁, u₂, u₃} is an orthogonal set, where

u₁ = , , 

13. Let , where  and  . Construct an orthogonal basis {v₁, v₂} for W.

14. Let * be defined on  by . Then show that  forms a group.

15. Define ring with an example. Compute the product in the given ring (12)(16) in .