Mathematics II 2070

Attempt all questions:

Group A (10 x 2 = 20)

1. Why the system x₁ - 3x₂ = 4; -3x₁ + 9x₂ = 8 is consistent? Give the graphical representation?

2. Define linear combination of vectors. If v₁, v₂,v₃ are vectors, Write the linear combination of 3v₁ - 5v₂ + 7v₃ as a matrix times a vector.

3. Is  invertible matrix?

4. Define invertible linear 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 vector space.

7. Show that the entries in the vector x = (1, 6) are the coordinates of x relative to the standard basis (e₁, e₂).

8. Is an Eigen value of  

9. Find the inner product of (1, 2, 3) and (2, 3, 4).

10. Compute the norm between the vectors 4 = (7, 1) and v = (3, 2).

Group B (5 x 4 = 20)

11. A linear transformation  is defined by . Find the image of T of  

12. If  compute (Ax)^T, x^TA^T and xx^T. Can you compute x^TA^T? 

13. If b₁ = (2, 1), and B = {b₁, b₂}, find the co-ordinate vector [x]_B of x relative to B.

14. Find the eigen values of .

15. Show that {v₁, v₂, v₃} is an orthogonal set, where v₁ = (3,1,1), v₂ = (-1, 2, 1), 

Group C (5 x 8 = 40)

16. Let a₁ = (1, 2, -5), a₂ = (2,5,-3) and b = (7,4, -3). Determine whether b can be generated as a linear combination of a₁ and a₂. That is, determine whether x₁ and x₂ exists such that x₁a₁ + x₂a₂ = b

has the solution , find it.

OR

Determine if the following system is consistent

x₂ - 4x₃ = 8

2x₁ - 3x₂ + 2x₃ = 1

5x₁ - 8x₂ + 7x₃ = 1

17. Compute the multiplication of partitioned matrices for 

18. Let b₁ = (1,0,3), b₂ = (1,-1,2) and x = (3,-5,4). Does B={b1, b2, b3} form a basis? Find [x]_B, for x.

19. Diagonalize the matrix, if possible 

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

OR

Find a least square solution of Ax = b for