Mathematics II 2071

Attempt all questions:

Group A (10 x 2 = 20)

1. What is a system of linear equations? When the system is consistent and inconsistent?

2. Define linearly dependent and independent vectors. If (1, 2) and (3, 6) are vectors then the vectors are linearly dependent or independent?

3. Define invertible matrix transformation. 

4. 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 

5. Show that the matrices do not commute.

6. Define vector space.

7. Determine if w = (1, 3, -4) is a Nul A, where 

8. Is u = (3, -2) is an eigen value of  ?

9. Find the inner product of (2, -5, -1) and (3, 2, -3).

10. Find the norm between the vectors u = (1, 2, 3, 4) and v = (0, 1, 2, 3).

Group B (5 X 4 = 20)

11. Let , u = (1, 0, -3) and v = (5, -1, 4), If   defined by T(x) = Ax, find T (u) and T (v).

13. If v₁ and v₂ are the vectors of a vector space V and H = span {v₁, v₂}, then show that H is a subspace of V.

14. Find the eigen values of  

15. Show that (v₁, v₂, v₃) is an orthogonal basis of R³, where 

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 inconsistent.

x₂ - 4x₃ = 8

2x₁ - 3x₂ + 2x₁ = 1

5x₁ - 8x₂ + 7x₃ = 1

OR

Let a₁ = (1, -2, -5), a₂ = (2, 5, 6) and b = (7, 4, -3) are the vectors. Determine whether b can be generated as a linear combination of a₁ and a₂. That is determine whether x₁ and x₂ exist such that x₁a₁ + x₂a₂ = b has solution, find it.

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.

OR

Compute the multiplication of partitioned matrices for

18. Let b₁ = (1, 0, 0), b₂ = (-3, 4, 0), b₃ = (3, -6, 3) and x = (-8, 2, 3) then 

(a) Show that B = {b₁, b₂, b₃} is a basis of R³.

(b) Find the change of co-ordinates matrix from B to the standard basis.

(c) Find [X]_B, for the given x.

19. Diagonalize the matrix, if possible

20. What is a least-squares solution? Find a least-squares solution of Ax = b, where