Institute of Science and Technology
Bachelor Level / second-semester / Science
Computer Science and Information Technology( MTH163 )
Full Marks: 80
Pass Marks: 32
Time: 3 Hours
Candidates are required to give their answers in their own words as far as practicable.
The figures in the margin indicate full marks.
Attempt any three questions.
Define system of linear equations. When a system of equations is consistent? Make echelon form to solve:
-2a – 3b + 4c = 5
b – 2c = 4
a + 3b – c = 2
Define linear transformation with an example.
Let A = , v = , b= , x =,
and define a transformation T : R² → R² by T(x) = Ax then
a. find T(v).
b. Find x ∈ R² whose image under T is b.
Find AB by block multiplication of the matrices.
Find the least square solution of Ax=c where
and compute the associated least square error.
Attempt any ten questions.
Determine the column of the matrix A are linearly independent where
Let A = and B = . What value (s) of k, if any, will make AB=BA?
Evaluate the determinant of the matrix.
When two column vectors in R² are equal? Give an example. Compute u+3v, -u-2v where,
Prove that the two vectors u and v are perpendicular to each other if and only if the line through u is perpendicular bisector of the line segment from -u to v.
Find the eigenvalue of A =
Define null space of a matrix A. Let
then show that v belongs to the null space matrix A.
Find the equation y = a0 + a1 x of the least squares line that best fits the data points (2,1), (5,2), (7,3), (8,3).
Show that the solution of yk+2 – 4yk+1 + 3yk = 0 are linearly independent.
Define group. Show that the set of integers is a group with respect to addition operation.