Questions
Syllabus
Qns. By Syllabus

Semester

Subject

Year

Tribhuwan University

Institute of Science and Technology

2076

Bachelor Level / First Year / Second Semester / Science

B.Sc in Computer Science and Information Technology (MTH168)

(Mathematics II)

Full Marks: 60

Pass Marks: 24

Time: 3 Hours

Candidates are required to give their answers in their own words as for as practicable.

The figures in the margin indicate full marks.

Section A

Long Answers Questions

Attempt any TWO questions.
[2*10=20]
1.
When a system of linear equation is consistent and inconsistent? Give an example for each. Test the consistency and solve:

$x - 2y = 5$

$-x + y + 5z = 2$

$y + z = 0$
[10]
2.
What is the condition of a matrix to have an inverse? Find the inverse of the matrix if it exists.

$A = \begin{bmatrix} 5 & 1 & 2 \\ 1 & 0 & 3 \\ 4 & -3 & 8 \end{bmatrix}$
[10]
3.
Find the least-square solution of Ax = b for

$$A = \begin{bmatrix} 1 & -6 \\ 1 & -2 \\ 1 & 1 \\ 1 & 7 \end{bmatrix}$ and $b = \begin{bmatrix} -1 \\ 2 \\ 1 \\ 6 \end{bmatrix}$$
[10]
4.
Let T is a linear transformation. Find the standard matrix of T such that:

$T: \mathbb{R}^2 \to \mathbb{R}^4 \text{ by } T(\mathbf{e}_1) = (3,1,3,1),\ T(\mathbf{e}_2) = (-5,2,0,0) \text{ where } \mathbf{e}_1 = (1,0),\ \mathbf{e}_2 = (0,1)$

$T: \mathbb{R}^2 \to \mathbb{R}^4 \text{ rotates points about the origin through } \frac{3\pi}{4} \text{ radians counter clockwise}$

$T: \mathbb{R}^2 \to \mathbb{R}^4 \text{ is a vertical shear transformation that maps } \mathbf{e}_1 \text{ into } \mathbf{e}_1 - 2\mathbf{e}_2 \text{ but leaves } \mathbf{e}_2 \text{ unchanged}$
[10]
Section B

Short Answers Questions

Attempt any Eight questions.
[8*5=40]
5.
For what value of h will y be in span ${ v_1, v_2, v_3 } $ if $v_1, v_2, v_3$ and y are given as:

$v_1 = \begin{bmatrix} 1 \\ -1 \\ -2 \end{bmatrix},\ v_2 = \begin{bmatrix} 5 \\ -4 \\ -7 \end{bmatrix},\ v_3 = \begin{bmatrix} -3 \\ 1 \\ 0 \end{bmatrix},\ \text{and } y = \begin{bmatrix} -4 \\ 3 \\ h \end{bmatrix}$
[5]
6.
Let us define a linear transformation $T: \mathbb{R}^2 \to \mathbb{R}^2$ by $T(x) = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} -x_2 \\ x_1 \end{bmatrix}$. Find the image under $T$ of $u = \begin{bmatrix} 4 \\ 1 \end{bmatrix}$, $v = \begin{bmatrix} 2 \\ 3 \end{bmatrix}$ and $u + v = \begin{bmatrix} 6 \\ 4 \end{bmatrix}$. [5]
7.
Let A and B be matrices. Determine the value of (s) of k if any will make AB = BA.

$A = \begin{bmatrix} 2 & 5 \\ -3 & 1 \end{bmatrix},\ B = \begin{bmatrix} 4 & -5 \\ 3 & k \end{bmatrix}$
[5]
8.
Define determinant. Compute the determinant without expanding:

$\begin{bmatrix} -2 & 8 & -9 \\ -1 & 7 & 0 \\ 1 & -4 & 2 \end{bmatrix}$
[5]
9.
Define null space. Find their basis for the null space of the matrix

$A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \end{bmatrix}$
[5]
10.
Let $B = \{b_1, b_2\}$ and $C = \{c_1, c_2\}$ be bases for a vector $V$, and suppose $b_1 = -c_1 + 4c_2$ and $b_2 = 5c_1 - 3c_2$. Find the change of coordinate matrix for a vector space and find $[x]_C$ for $x = 5b_1 + 3b_2$. [5]
11.
Find the eigen values of the matrix

$\begin{bmatrix} 6 & 5 \\ -8 & -6 \end{bmatrix}$
[5]
12.
Find the QR factorization of the matrix

$\begin{bmatrix} 2 & 1 \\ 3 & -1 \end{bmatrix}$
[5]
13.
Define binary operation. Determine whether the binary operation Q is associative or commutative or both where Q is defined on Q by letting .

$x * y = \frac{x + y}{3}$
[5]
14.
Show that the ring ($Z_4$, + 4, 4) is an integral domain. [5]
15.
Find the vector $x$ determined by the coordinate vector $[x]_\beta = \begin{bmatrix} -4 \\ 8 \\ 7 \end{bmatrix}$ where $\beta = \left\{ \begin{bmatrix} -1 \\ 2 \\ 0 \end{bmatrix}, \begin{bmatrix} 3 \\ -5 \\ 2 \end{bmatrix}, \begin{bmatrix} 4 \\ -7 \\ 3 \end{bmatrix} \right\}$. [5]