Linear Algebra (MTH-155)

          

Pre-Semester Examination-2070            

Set-A

B.Sc. CSIT/ 2nd Semester/2069 batch

Course: Linear Algebra (MTH-155)                                                                                Full Marks: 80

Pass Marks: 32   Time: 3hrs     

Attempt all the questions.

                                    

Group-A                                                                                                                       10 2=20

              

1.      For what value of h and k is the following system consistent 2x₁ – x₂=h, -6x₁ + 3x₂ =k.

2.      Verify that  is an Eigen vector of A=

3.      Define singular and non singular matrices.

4.      Find the area of parallelogram whose vertices are (0,0),(5,2),(6,4) and (11,6)

5.      Determine if u=  is in Null(A), where A=

6.      State the numerical importance of determinant calculation by row operation.

7.      Consider a basis B = {b₁,b₂} for IR² , where b₁= , b₂= .find x  IR²  so that co-ordinate vector [x]_b=

8.      Let T: IR² IR²  be a linear transformation defined by T(x ,y) = (x+y , x-y). find Ker(T)

9.      If   is an Eigen value of A. Find the Eigen value of A^-1 .

10.  Let w= span{x_{1 ,} x₂}, where x_{1 =    ,} x₂=    then construct an orthogonal basis for w.

Group-B                                                                                                                           5 4=20

11.  Let T : IR³ IR³ be the linear transformation defined by T(x , y , z) = (x , y , x-2y) .Find basis and dimension of (a) Ker(T)                                                                                                                                                  (b)I_m(T)

12.  Let u and v be non zero vectors in IRⁿ  and the angle between them is . Prove that

u .v =   cos

            OR

State and prove orthogonality property for any two non zero vectors in IRⁿ .

13.  Given the Leontief input output model x = cx + d , where symbols have their usual meanings. Consider

Any economy whose consumption matrix is given by c = . Suppose the final demand is 50 units for manufacturing, 30 units for agriculture and 20 units for services. Find the production level x that will satisfy this demand.

14.  Determine the Eigen values and Eigen Vectors of A=  

15.  Find the matrix representation of Linear transformation T: IR²  IR² defined by T(x, y) = (x, x+2y) relative to standard basis.

Group-C                                                                                                                          5 8= 40

16. Find a matrix A whose inverse is A-1 =   using elementary row reduces the augmented matrix.

17. Verify that the set of all matrices of the form  is a subspace of vector space of all 33 matrices

OR

Define vector space, subspace, and basis of a vector space with suitable example. What do you mean by linearly dependent and linearly independent set of vectors?

18. Determine if the following homogeneous system has a non trivial solution. then describe the solution set 3x1 + 5x2 – 4x3=0 , -3x1 - 2x2 + 4x3=0 , 6x1 + x2 – 8x3=0

19. Verify Cayley Hamilton theorem for matrix A = 

OR

Diagonalize the matrix A=   , if possible.

20. What do you mean by least square lines? find the equation y = 0 + 1x  of the least square line that fits the data points(2,0),(3,4),(4,10),(5,16).

Mid Term Examination-2070            

Set-B

B.Sc. CSIT/ 2nd Semester/2069 batch

Course: Linear Algebra (MTH-155)                                                                                Full Marks: 80

Pass Marks: 32   Time: 3hrs      

Attempt all the questions.

Group-A                                                                                                                      10    

1.      When is system of linear equation consistent or inconsistent?

2.      How do you distinguish singular and non singular matrices?

3.      When is a linear transformation invertible?

4.      If A and B are n  n matrices, then verify that det(AB) = det(A) . det(B)

5.      Calculate the area of parallelogram determined by the columns of A= 

6.      Find a matrix A such that W = col(A) if W= .

7.      Show that {(1, 1), (-1, 0)} forms a basis for IR² .

8.      Show that 7 is an Eigen value of A = 

9.      Find the distance between vectors u= (7, 1) and v= (3, 2). Define the distance between two vectors in IRⁿ.

10. Let W=span{x₁ , x₂}, where x₁ = , x₂ = . Then construct orthogonal basis for W.

Group-B                                                                                                          5    

11. Determine if  the following  set of vectors are linearly dependent or independent (1,1,2),(3,1,2),(0,1,4).

12. Find a 33 matrix that corresponds to the composite transformation of a scaling by 0.3, a rotation of 90 and finally a translation that adds (-0.5, 2) to the each point of a figure.                                                                                                                                                      OR                                                                                                                                                         

       Let T : IR³ IR³ be the linear transformation defined by T(x , y , z) = (x , y , x-2y) .Find a basis and dimension of (a) Ker(T)                                                                                                                                                          (b)I_m(T)

13. In the vector space IR² , express the vector x=(1 , 2) as a Linear combination of vectors in a basis

B = {(1,-1), (0, 1)}. Also write [x]­_B

14. When does a matrix A is said to be similar to B? Prove that, if n  n matrices A and B are similar, then they have same characteristics polynomial and hence same Eigen values.

15. Let u and v are non zero vectors in IR2,then prove that angle  between them is given by

u .v =   cos.

Group-C                                                                                                               5    

16. Define basis of a vector space. Find the basis for the row space of  A =  

17. Let IRⁿ IR^m  be a  linear transformation. Then prove that T is one to one if and only if the equation T(x) = 0 has only the trivial solution.

Or

Let A =  , u=  , b=  , c=  and define  T : IR² IR³ by T(x) = Ax, then

(a)  Find T (u)

(b)  Find an x  IR² whose image under T is b.

(c)  Is there more than one x whose image under T is b.

(d)  Determine if c is in the range of T.

18. Find the inverse of    if it exists, by using elementary row reduce the augmented matrix.

19. Diagonalize the matrix   , if possible.

20. Find the least square of Ax = b if A =    , b=  

OR

What do you mean by least square lines? Find the equation y= 0 + 1x  of the least square line that fits the data points(2,1),(5,2),(7,3),(8,3).