Mathematics II 2069

Attempt all questions:

Group A (10 x 2 = 20)

1. What do you mean by linearly independent set and linearly dependent set of vectors?

3. What do you mean by consistent equations? Give suitable examples.

4. What do you mean by change of basis in Rⁿ?

5. Find the dimension of the vector spanned by (1,1,0) and (0,1,0).

6. When is a linear transformation invertible.

7. Find the rank of AB where 

8. Is  an Eigen value of  ?

9. Define kernel and image of linear transformation.

10. What is meant by Discrete dynamical system? Give suitable example.

Group B (5 x 4 = 20)

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

12. Show that the following vectors are linearly independent:(1, 1, 2),(3, 1, 2),(0, 1, 4).

13. Find the matrix representation of linear transformation defined by T(x, y) = (x + 2y) relative to the standard basis.

14. Is the set of vectors {(91, 0, 1),(0, 1, 0),(-1, 0, 1)} orthogonal? Obtain the corresponding orthonomal set R³.

15. Let the four vertices O(0, 0), A(1, 0), B(0, 1) and C(1, 1) of a unit square be represented by 2 x 4 matrix . Investigate and interpret geometrically the effect of pre-multiplication of this matric by the 2 x 2 matrix:

OR

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

Group C (5 x 8 = 40)

16. Find a matrix A whose inverse is 

17. Test the consistency and solve

x + y + z = 4

x + 2y + 2z = 2

2x + 2y + z = 5

OR

Verify Cayley Hamilton theorem for matrix 

18. The set of matrices of the form 

is a subspace of the vector 3 x 3 matrices. Verify it.

19. Let V and W be the vector spaces over a field F of real numbers. Let dim V = n and dim W = m. Let {e₁,e₂,... ...,e_n} be a basis of V and {f₁, f₂, ... ... ... , f_m} be a basis of W. Then, prove that each linear transformation can be represented by an m x n matrix A with elements from F such that 

Y = AX

Are column matrices of coordinates of relative to its basis and coordinates of relative to its basis respectively.

OR

Compute the multiplication partitioned matrices for 

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