Mathematics II 2072

Attempt all questions:

Group A (10 x 2 = 20)

1. Define linear combination of vectors. When the vectors are linearly dependent and independent?

2. Define linear transformation between two vector spaces.

3. Show that the matrix is not invertible.

4. Define invertible matrix transformation.

5. Let S be the parallelogram determined by the vectors b₁ = (1,3) and b₂ = (5,1) and let . Compute the area of the image S under the mapping 

6. Define subspace of a vector space.

7. If and let u = (5, 3, 2), then show that u is in the Nul A.

8. If u = (6, -5) is an eigen vector of ?

9. Find the unit vector u of v = (1, -2, 2, 0) along the direction of v.

10. Find the norm vector  v = (1, -2, 3, 0).

Group B (5 x 4 = 20)

11. Let  and  . Find the images under T of 

12. Find the determinant of 

13. Show that the vectors (1, 0, 0),(1, 1, 0) and (1, 1, 1) are linearly independent.

14. Find the eigen values of 

15. If v₁ = (3, 6, 0), v₂ = (0, 0, 2) are the orthogonal basis then find the orthonal basis of v₁ and v₂.

OR

Find an orthogonal projection of y onto u, where y = (7, 6), u = (4, 2)

Group C (5 x 8 = 40)

16. Determine if the following system is consistent, if consistent solve the system.

-2x₁ - 3x₂ + 4x₃ = 5

x₁ - 2x₂ = 4

x₁ + 3x₂ - x₃ = 2

OR

Let   and define a transformation   so that 

a) Find T(u)

b) Find x in R² whose image under T is b.

17. If the consumption matrix C is 

and 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.

18. Let v₁ = (3, 6, 2), v₂ = (-1, 0, 1), x = (3, 12, 7) and B = {v₁, v₂}. Then B is a basis for H = span {v₁, v₂}. Determine if x is in H, and if it is, find the co-ordinate vector of x relative to B. 

19. Diagonalize the matrix, if possible 

20. Find the equation y = a₀ + a₁x for the least squares line that best fits the data points (2, 1), (5, 2), (7, 3),(8, 3).

OR

When two vectors 4 and v are orthogonal? If u and vectors, prove that [dist (u, -v)]² = [dist (u, v)]² if u.v = 0.