# Mathematics II 2074

**Tribhuwan University**

**Institute of Science and Technology**

**2074**

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

Attempt all questions:

Group A (10 x 2 = 20)

1. What are the criteria for a rectangular matrix to be in echelon form?

2. prove that (a) (A^{T})^{T} = A (b) (A + B)^{T} = A^{T} + B^{T} , Where A and B denote matrices whose size are appropriate for the above mentioned operations.

4. Let A and B be two square matrices. By taking suitable examples, show that even though AB and BA may not be equal, it is always true that detAB = detBA.

5. Using Cramer's rule solve the following simultaneous equations:

5x + 7y = 3

2x + 4y = 1

6. Define vector space with suitable examples.

7. Let **W** be the set of all vectors of the form , where b and c are arbitrary. Find vector **u** and **v** such that **W** = Span {**u**, **v**}.

8. What are necessary and sufficient conditions for a matrix to be invertible?

9. Determine whether the pair of vectors are orthogonal or not?

10. What do you understand by least square line? Illustrate.

Group B (5 x 4 = 20)

11. What are the criteria for a transformation T to be linear? If is defined by T(x) = 3x,

Show that T is a linear transformation. Also give a geometric description of the transformation

12. Prove that if A is an invertible matrix, then so is A^{T}, and the inverse of A^{T} is the transpose of A^{-1}.

13. Define subspace of a vector space **V**. Given v_{1} and v_{2} in a vector space **V**, let H = span {v_{1}, v_{2}}. Show that **H** is a subspace of **V**.

**OR**

If is a basis for a vector space V and x is in V, define the coordinate of x relative to the basis . Let Then is a basis for H = Span {v_{1}, v_{2}}. Determine is X is in **H**, and if it is, find the coordinate vector of **x** relative to

14. The mapping defined by T(a_{0} + a_{1}t + a_{2}t^{2}) = a_{1} + 2a_{2}t is a linear transformation.

a) Find the matrix for T, when is the basis {1, t, t^{2}}.

b) Verify that [T(**p**)]_{B} = [T]_{B}[**p**]_{B} for each** p** in **P**_{2}.

15. Let Find a least square solution of Ax = b, and compute the associated least square error.

Group C (5 x 8 = 40)

16. Let be a linear transformation and let A be the standard matrix for T. Then prove that: T map **R ^{n} **on to

**R**if and only if the columns of A span

^{m}**R**; and T is one-to-one if and only if the columns of A are linearly independent. Let T(x

^{m}_{1}, x

_{2}) = (3x

_{1}+ x

_{2}, 5x

_{1}+ 7x

_{2}, x

_{1}+ 3x

_{2}). Show that T is a one-to-one linear transformation. Does T map

**R**onto

^{2}**R**?

^{3}17. Compute the multiplication of partitioned matrices for

18. Let be a basis for a vector space V. Then the coordinate mapping is one-to-one linear transformation from V into R^{n}. Let

a) Show that the set is a basis of **R ^{3}**.

b) Find the change of coordinates matrix for to the standard basis.

c) Write the equation that relates x in **R ^{3}** to .

d) Find for the x give above.

19. Diagonalize the matrix if possible.

**OR**

Suppose A = PDP^{-1}, where D is a diagonal n x n matrix. If is the basis for **R ^{n}** formed for the columns of P, then prove that D is the matrix for the transformation. Define , where Find a basis for

**R**with the property that the matrix for T is a diagonal matrix.

^{2}20. What is a least squares solution? Find a least squares solution of Ax = b, where

**OR**

What do you understand by orthonormal set? Show that {v_{1}, v_{2}, v_{3}} is an orthonormal basis of **R ^{3}**, where

Prove that an m x n matrix U has orthonormal columns if and only if U^{T}U = 1.