Mathematics II 2065

Attempt all questions:

Group A (10 x 2 = 20)

1. Illustrate by an example that a system of linear equations has either equations has either exactly one solution or infinitely many solutions.

2. When is a linear transformation invertible?

3. Solve the system 

3x₁ + 4x₂ = 3,               5x₁ + 6x₂ = 7

by using the inverse of the matrix 

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

5. State Cramer’s rule for an invertible n x n matrix A and vector  to solve the system Ax = b. Is this method efficient from computational point of view?

6. Determine if {v₁, v₂, v₃} is basis for R³, where    

7. Determine if  is a Nul(A) for 

8. Show that 7 is an eigen value of 

9. If S = {u₁,... .... ... ... , u_p} is an orthogonal set of nonzero vectors in R², show S is linearly independent and hence is a basis for the subspace spanned by S.

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

Group B (5 x 4 = 20)

11. Determine if the given set is linearly dependent:

12. Find the 3 x 3 matrix that corresponds to the composite transformation of a scaling by 0.3, a rotation of 90⁰ , and finally a translation that adds (-0.5, 2) to each point of a figure.

OR

Describe the Leontief Input-Output model for certain economy and derive formula for (I-C)-1, where

symbols have their usual meanings.

13. Find the coordinate vector [X]_B of a x relative to the given basis B = {b₁, b₂}, where

14. Let   and basis B = {b1, b2}.Find the B-matrix for the transformation   with P = {b₁, b₂}.

15. Let u and v be non-zero vectors in R³ and the angle between them be  Then prove that where the symbols have their usual meanings.

Group C (5 x 8 = 40)

16. Let   be a linear transformation. Then T is one-to-one if and only if the equation T(x) = 0

has only the trivial solution, prove the statement.

OR

Let  and define Then

a) Find T(u)

b) Find an whose image under T is b.

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

d) Determine if c is the range of T.

17. Compute the multiplication of partitioned matrices for 

18. What do you mean by change of basis in Rⁿ? Let and consider the bases for R² given by B = {b₁, b₂} and C = {c₁, c₂}.

a) Find the change of coordinate matrix from C to B.

b) Find the change of coordinate matrix from B to C.

OR

Define vector spaces, subspaces, basis of vector space with suitable examples. What do you mean by

linearly independent set and linearly dependent set of vectors?

19. Diagonalize the matrix  if possible.

20. Find the equation  of the least squares line that best fits the data points (2, 1), (5, 2),

(7, 3), (8, 3). What do you mean by least squares lines?