This document outlines the structure and content of the Fourth Semester Mathematics examination for Linear Algebra, held in April 2021. It includes sections for short answer questions, paragraph/problem type questions, and essay type questions, with specific marks allocated to each section. The questions cover various topics in linear algebra, including systems of linear equations, matrix operations, vector spaces, and eigenvalues.
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Mathematics
This document outlines the structure and content of the Fourth Semester Mathematics examination for Linear Algebra, held in April 2021. It includes sections for short answer questions, paragraph/problem type questions, and essay type questions, with specific marks allocated to each section. The questions cover various topics in linear algebra, including systems of linear equations, matrix operations, vector spaces, and eigenvalues.
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF or read online on Scribd
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C 3555 Le (Pages :4) Namen
: Reg. No.....
b FOURTH SEMESTER (CBCSS—UG) DEGREE EXAMINATION
A APRIL 2021
t
ee Mathematics
MTS 4B 04—LINEAR ALGEBRA
‘Time : Two Hours and. a Half Maximum : 80 Marks
Section A (Short Answer Type Questions)
Answer at least ten questions.
Each question carries 3 marks.
All questions can be attended.
Overall Ceiling 30.
1. Describe different possibilities for solution (x, y) of a system linear equations in the xy plane.
What are consistent system ?
2. Suppose that the augmented matrix for a linear system has been reduced to the row echelon form
1-347
as|° 1 2 2) solve the system.
0 015
a
z n 3
8. Define trace of a square matric. Find the trace of the matrix A=)
4
4. Show that the standard unit vectors
1 =(1, 0,....0), €2 = (0,1, 0....0), €3 = (0, 0, 1,0....0). -1)span R".
5. Find the co-ordinate vector of w=(1,0) relative to the basis
iy, Zp) of R®, where it = (1, ~ 1) and tz = (1, 1).
6. Write two important facts about the vectors in a finite dimensional vector space V. 2 C 3555,
7. Consider the bases B = [i, iz] and B' =| Al where
i =(1, 0), % =(0,1),% (2,1). Find the transition matrix Py’, 3 from B' to B.
Q, oe
8. Define row spaces and null spaces an m x n matrix.
is the row reduced echelon form of a 3 x 3 matrix A, then verify the rank-
nullity formula.
10. Show that the operator T:R® -» R? that rotates vectors through an angle @ is one-one.
5 2
11. Find the image of the line y = 4x under multiplication by the matrix A= i i}
12. Confirm by multiplication that x is an eigen vector of A and find the corresponding eigen value if
3 0 1
BJ
A. Let A be ann x n matrix. Define inner product on R" generated by A. Also write the generating
matrix of the weighted Euclidear inner product =wyi4by + Wg) + pe
+ Up!
@ If.u, v are vectors in a real inner product space V, then show that ||u+v|[<]u]+]» |.
15." IPA is ann xn orthogonal matrix, then show that | Az |=] x | for all xin R”.
(10 x 3 = 30 marks) 16.
17.
18.
19.
20.
21.
22.
@
3 C3555,
Section B (Paragraph/Problem Type Questions)
Answer at least five questions.
Each question carries 6 marks.
All questions can be attended.
Overall Ceiling 30.
Describe Column Row Expansion method for finding the product AB for two matrices A and B. Use
; 1 3if 204
this to find the product AB = .
e product AB IF Al : ‘|
of
and v=(6,4,2) in R®. Show that w=(9,2,7) is a linear
If A is an invertible matrix, then show that AT is also invertible and (ry
Consider the vectors u=(1,2,-1)
combination of u and v and that w’ =(4,~1,8) is not a linear combination of u and v.
is a basis for a vector space V, then show that every vector v in V can be
If = {0,02
expressed in form v=C)-+6.%2 to +é,0q in exactly one way. What are the co-ordinates of v
relative to the basis s.
ix with n columns, then detine rank of A and show that rank (A) + nullity (A) =.
IfA is a mati
Find the standard matrix for the operator T: R® —, R? that first rotates a vector counter clockwise
about z-axis through an angle 0, then reflects the resulting veetor about yz plane and then projects
that vector orthogonally onto the xy plane.
Define eigen space corresponding to an eigen value 2 of square matrix A. Also find eigen rine
Z 1 +2
and bases for the eigen space of the matrix A = E ‘I
fw is a sub-space of real inner produet space», then show that:
(a) wt is subspace of v.
() wow" ={0}-
(6 x 6 = 30 marks)
Turn over 24.
25.
26.
27.
4 C3555 /
Section C (Essay Type Questions)
Answer any two questions.
Each question carries 10 marks.
(a) Show that every elementary matrix is invertible and the inverse is also an elementary matrix.
oan
12 3
(b) Find the inverse of A=|2 5 3| using Row operations.
108
(a) Let V be a vector space and qj a vector in V and K a scalar. Then show that :
(a) 07=0 ;and
(b) (-1)a=
(b) Show that the vectors v, = (1, 2,1), v2 = (2, 9, 0) and vs =(3, 3, 4) form a basis for R°.
(a) Consider the basis B=[4,ug]and B'= [1;",u:'] for R? where um =(2,2), tug =(4,~1)
uy =(18)uy = (4-2)
(i) Find the transition matrix B' toB.
(ii) Find the transition matrix B to B.
(b) Find the reflection of the vector x=(1,5) about the line through the origin that makes an
angle of with the x-axis.
‘When you can say that a square matrix A is diagonalizable ? If A is an n x n matrix, show that the
following statements are equivalent :
(a) Ais diagonalizable ; and
(b) Ahas n linearly independent eigen vectors.
(2x 10 = 20 marks)
Regular Arrear Examination - December 2013 Course: MAT202 - Linear Algebra Time: Three Hours Max - Marks:100 PART - A (10 X 3 30 Marks) Answer ALL Questions
