# Mathematics II 2070

**Tribhuwan University**

**Institute of Science and Technology**

**2070**

**Computer Science and Information Technology ( MTH163 )**

Attempt all questions:

Group A (10 x 2 = 20)

1. Why the system x_{1} - 3x_{2} = 4; -3x_{1} + 9x_{2} = 8 is consistent? Give the graphical representation?

2. Define linear combination of vectors. If v_{1}, v_{2},v_{3} are vectors, Write the linear combination of 3v_{1} - 5v_{2} + 7v_{3} 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 (e_{1}, e_{2}).

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^{T}A^{T} and xx^{T}. Can you compute x^{T}A^{T}?

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

14. Find the eigen values of .

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

Group C (5 x 8 = 40)

16. Let a_{1} = (1, 2, -5), a_{2} = (2,5,-3) and b = (7,4, -3). Determine whether b can be generated as a linear combination of a_{1} and a_{2}. That is, determine whether x_{1} and x_{2} exists such that x_{1}a_{1} + x_{2}a_{2} = b

has the solution , find it.

**OR**

Determine if the following system is consistent

x_{2} - 4x_{3} = 8

2x_{1} - 3x_{2} + 2x_{3 }= 1

5x_{1} - 8x_{2} + 7x_{3} = 1

17. Compute the multiplication of partitioned matrices for

18. Let b_{1} = (1,0,3), b_{2} = (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