Mathematics II 2075(New Course)

Group A 

Attempt any three questions:(3 x 10 = 30)

1. When a system of linear equation is consistent and inconsistent? Give an example for each. Test the consistency and solve: x + y + z = 4, x + 2y + 2z = 2, 2x + 2y + z = 5.

2. What is the condition of a matrix to have an inverse? Find the inverse of the matrix in exists.

3. Define linearly independent set of vectors with an example. Show that the vectors (1, 4, 3), (0, 3, 1) and (3, -5, 4) are linearly independent. Do they form a basis? Justify.

4. Find the least-square solution of Ax = b for 

Attempt any ten questions: (5 x 10 = 50)

Group B 

5. Change into reduce echelon form of the matrix 

6. Define linear transformation with an example. Is a transformation defined by T(x, y) = (3x + y, 5x + 7y, x + 3y) linear? Justify.

7. Let What value (s) of k if any will make AB = BA?

8. Define determinant. Evaluate without expanding 

9. Define subspace of a vector space. Let Show that H is a subspace of:

10. Find the dimension of the null space and column space of 

11. Find the eigenvalues of the matrix 

12. Find LU factorization of the matrix 

13. Define group. Show that the set of all integers Z forms group under addition operation.

14. Define ring with an example. Compute the product in the given ring (-3, 5) (2, -4) in Z₄ x Z₁₁.

15. State and prove the Pythagorean theorem of two vectors and verify this for u = (1, -1) and v = (1, 1).