Questions
Syllabus
Qns. By Syllabus

Semester

Subject

Year

Tribhuwan University

Institute of Science and Technology

2080.1

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.
What is a system of linear equations? When the system is consistent? Find the condition on g, h, k that makes the system consistent.
x1−4x2+7x3=gx_1 - 4x_2 + 7x_3 = g
3x2−5x3=h3x_2 - 5x_3 = h
−2x1+5x2−9x3=k-2x_1 + 5x_2 - 9x_3 = k
[10]
2.
define a transformation T:R3→R2 by T(x)=AxT : \mathbb{R}^3 \to \mathbb{R}^2 \text{ by } T(x) = Ax then
letA=[1−5−7−375], u=[123], b=[−2−2], T(x)=Axlet A = \begin{bmatrix} 1 & -5 & -7 \\ -3 & 7 & 5 \end{bmatrix},\ u = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix},\ b = \begin{bmatrix} -2 \\ -2 \end{bmatrix},\ T(x) = Ax
a.find T(u)a. \text{find } T(u)
b.Find x∈R3 whose image under T is bb. \text{Find } x \in \mathbb{R}^3 \text{ whose image under } T \text{ is } b
c.Is x unique?c. \text{Is } x \text{ unique?}
[10]
3.
Find the least square solution of Ax = b where and compute the associated least square error.
A=[1−3−3151172], b=[5−3−5]A = \begin{bmatrix} 1 & -3 & -3 \\ 1 & 5 & 1 \\ 1 & 7 & 2 \end{bmatrix},\ b = \begin{bmatrix} 5 \\ -3 \\ -5 \end{bmatrix}
[10]
Section B

Short Answers Questions

Attempt any Eight questions.
[8*5=40]
4.
Are vectors v1v_1, v2v_2, and v3v_3 linearly independent? Justify.
v1=[140], v2=[1021], v3=[−506]v_1 = \begin{bmatrix} 1 \\ 4 \\ 0 \end{bmatrix},\ v_2 = \begin{bmatrix} 10 \\ 2 \\ 1 \end{bmatrix},\ v_3 = \begin{bmatrix} -5 \\ 0 \\ 6 \end{bmatrix}
[5]
5.
Find LU Factorization. Given the matrix:
[2344510482]\begin{bmatrix} 2 & 3 & 4 \\ 4 & 5 & 10 \\ 4 & 8 & 2 \end{bmatrix}
[5]
6.
Compute Det of A where
A=[2−8683−9510−301−21−406]A = \begin{bmatrix} 2 & -8 & 6 & 8 \\ 3 & -9 & 5 & 10 \\ -3 & 0 & 1 & -2 \\ 1 & -4 & 0 & 6 \end{bmatrix}
[5]
7.
Show that H={(a−3b,b−a,a,b):a,b∈R}H = \{(a-3b, b-a, a, b) : a, b \in \mathbb{R}\} is a subspace of R4\mathbb{R}^4. [5]
8.
Is [32]\begin{bmatrix} 3 \\ 2 \end{bmatrix} an eigen vector of [5−3−49]\begin{bmatrix} 5 & -3 \\ -4 & 9 \end{bmatrix} ? If so, find eigenvalue. [5]
9.
Let u = (1, -2, 2, 0). Find a unit vector of v in the same direction of u. [5]
10.
Find the basis and dimension of Null A where A =
A=[12342478]A = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 2 & 4 & 7 & 8 \end{bmatrix}
[5]
11.
Define group. Show that (Z, .) doesn't form a group. [5]
12.
Show that every field is an integral domain. [5]