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 
.