Questions
Syllabus
Qns. By Syllabus

Semester

Subject

Year

Bachelors Level/First Year/Second Semester/Science csit/second semester/mathematics ii/syllabus wise questions

B.Sc Computer Science and Information Technology

Institute of Science and Technology, TU

Mathematics II (MTH168)

Year Asked: 2076, syllabus wise question

Determinants
1.
Define determinant. Compute the determinant without expanding:
[−28−9−1701−42]\begin{bmatrix} -2 & 8 & -9 \\ -1 & 7 & 0 \\ 1 & -4 & 2 \end{bmatrix}
[5]
Eigenvalues and Eigen Vectors
1.
Find the eigen values of the matrix
[65−8−6]\begin{bmatrix} 6 & 5 \\ -8 & -6 \end{bmatrix}
[5]
Linear Equations in Linear Algebra
1.
When a system of linear equation is consistent and inconsistent? Give an example for each. Test the consistency and solve:
x−2y=5x - 2y = 5
−x+y+5z=2-x + y + 5z = 2
y+z=0y + z = 0
[10]
Matrix Algebra
1.
What is the condition of a matrix to have an inverse? Find the inverse of the matrix if it exists.
A=[5121034−38]A = \begin{bmatrix} 5 & 1 & 2 \\ 1 & 0 & 3 \\ 4 & -3 & 8 \end{bmatrix}
[10]
2.
Let A and B be matrices. Determine the value of (s) of k if any will make AB = BA.
A=[25−31], B=[4−53k]A = \begin{bmatrix} 2 & 5 \\ -3 & 1 \end{bmatrix},\ B = \begin{bmatrix} 4 & -5 \\ 3 & k \end{bmatrix}
[5]
3.
Find the QR factorization of the matrix
[213−1]\begin{bmatrix} 2 & 1 \\ 3 & -1 \end{bmatrix}
[5]
Orthogonality and Least Squares
1.
Find the least-square solution of Ax = b for
A=[1−61−21117]andb=[−1216]A = \begin{bmatrix} 1 & -6 \\ 1 & -2 \\ 1 & 1 \\ 1 & 7 \end{bmatrix} and b = \begin{bmatrix} -1 \\ 2 \\ 1 \\ 6 \end{bmatrix}
[10]
Rings and Fields
1.
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=x+y3x * y = \frac{x + y}{3}
[5]
2.
Show that the ring (Z4Z_4, + 4, 4) is an integral domain. [5]
Transformation
1.
Let T is a linear transformation. Find the standard matrix of T such that:
T:R2→R4 by T(e1)=(3,1,3,1), T(e2)=(−5,2,0,0) where e1=(1,0), e2=(0,1)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:R2→R4 rotates points about the origin through 3π4 radians counter clockwiseT: \mathbb{R}^2 \to \mathbb{R}^4 \text{ rotates points about the origin through } \frac{3\pi}{4} \text{ radians counter clockwise}
T:R2→R4 is a vertical shear transformation that maps e1 into e1−2e2 but leaves e2 unchangedT: \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]
2.
Let us define a linear transformation T:R2→R2T: \mathbb{R}^2 \to \mathbb{R}^2 by T(x)=[0−110][x1x2]=[−x2x1]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 TT of u=[41]u = \begin{bmatrix} 4 \\ 1 \end{bmatrix}, v=[23]v = \begin{bmatrix} 2 \\ 3 \end{bmatrix} and u+v=[64]u + v = \begin{bmatrix} 6 \\ 4 \end{bmatrix}. [5]
Vector Space Continued
1.
Define null space. Find their basis for the null space of the matrix
A=[123234]A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \end{bmatrix}
[5]
2.
Let B={b1,b2}B = \{b_1, b_2\} and C={c1,c2}C = \{c_1, c_2\} be bases for a vector VV, and suppose b1=−c1+4c2b_1 = -c_1 + 4c_2 and b2=5c1−3c2b_2 = 5c_1 - 3c_2. Find the change of coordinate matrix for a vector space and find [x]C[x]_C for x=5b1+3b2x = 5b_1 + 3b_2. [5]
Vector Spaces
1.
For what value of h will y be in span v1,v2,v3{ v_1, v_2, v_3 } if v1,v2,v3v_1, v_2, v_3 and y are given as:
v1=[1−1−2], v2=[5−4−7], v3=[−310], and y=[−43h]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]
2.
Find the vector xx determined by the coordinate vector [x]β=[−487][x]_\beta = \begin{bmatrix} -4 \\ 8 \\ 7 \end{bmatrix} where β={[−120],[3−52],[4−73]}\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]