# Mathematics II 2065

**Tribhuwan University**

**Institute of Science and Technology**

**2065**

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

**Attempt all questions:**

**Group A (10 x 2 = 20)**

1. Illustrate by an example that a system of linear equations has either equations has either exactly one solution or infinitely many solutions.

2. When is a linear transformation invertible?

3. Solve the system

3x_{1} + 4x_{2} = 3, 5x_{1} + 6x_{2} = 7

by using the inverse of the matrix

4. State the numerical importance of determinant calculation by row operation.

5. State Cramerâ€™s rule for an invertible n x n matrix A and vector to solve the system Ax = b. Is this method efficient from computational point of view?

6. Determine if {v_{1}, v_{2}, v_{3}} is basis for R^{3}, where

7. Determine if is a Nul(A) for

8. Show that 7 is an eigen value of

9. If S = {u_{1},... .... ... ... , u_{p}} is an orthogonal set of nonzero vectors in R^{2}, show S is linearly independent and hence is a basis for the subspace spanned by S.

10. Let W = span{x1, x2} where and Their construct orthogonal basis for W.

**Group B (5 x 4 = 20)**

11. Determine if the given set is linearly dependent:

12. Find the 3 x 3 matrix that corresponds to the composite transformation of a scaling by 0.3, a rotation of 90^{0} , and finally a translation that adds (-0.5, 2) to each point of a figure.

**OR**

Describe the Leontief Input-Output model for certain economy and derive formula for (I-C)-1, where

symbols have their usual meanings.

13. Find the coordinate vector [X]_{B} of a x relative to the given basis B = {b_{1}, b_{2}}, where

14. Let and basis B = {b1, b2}.Find the B-matrix for the transformation with P = {b_{1}, b_{2}}.

15. Let u and v be non-zero vectors in R^{3} and the angle between them be Then prove that where the symbols have their usual meanings.

**Group C (5 x 8 = 40)**

16. Let be a linear transformation. Then T is one-to-one if and only if the equation T(x) = 0

has only the trivial solution, prove the statement.

**OR**

Let and define Then

a) Find T(u)

b) Find an whose image under T is b.

c) Is there more than one x whose image under T is b?

d) Determine if c is the range of T.

17. Compute the multiplication of partitioned matrices for

18. What do you mean by change of basis in R^{n}? Let and consider the bases for R^{2 }given by B = {b_{1}, b_{2}} and C = {c_{1}, c_{2}}.

a) Find the change of coordinate matrix from C to B.

b) Find the change of coordinate matrix from B to C.

**OR**

Define vector spaces, subspaces, basis of vector space with suitable examples. What do you mean by

linearly independent set and linearly dependent set of vectors?

19. Diagonalize the matrix if possible.

20. Find the equation of the least squares line that best fits the data points (2, 1), (5, 2),

(7, 3), (8, 3). What do you mean by least squares lines?