This document contains instructions for a linear algebra exam with 3 questions. Question 1 asks students to find a basis for a subspace spanned by a given set of polynomials. Question 2 asks students to find the transition matrix from one basis to another basis given the coordinate representations of the basis vectors. Question 3 asks students to determine if a given set of pairs of real numbers forms a vector space and identify any vector space axioms that fail.

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COMSATS University Islamabad, Lahore Campus
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Sessional -II
Course
Title:
Linear Algebra Course Code: MTH-231 Credit
Hours:
3(2,1)
Course
Instructor/s:
Dr. Mohsan Hassan Programme Name: BPH
Semester: Batch:FA16 Section: A/B Date: 03-05-2021
Time
Allowed:
90 Minutes Maximum Marks: 30
Student’s
Name:
Reg. No.
Important Instructions / Guidelines:
• Attempt all questions.
Q.1 Find a basis for the subspace of 3
P spanned by
 
3 2 3 2 3
2 1, 3 1, 2, 1, 1
S t t t t t t t t t
= + + + − + + + + +
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Q. 2: Consider the bases  
1 2
,
B u u
= and  
1 2
' ,
B w w
=
1 2 1 2
2 4 1 1
, , ,
2 1 3 1
u u w w
−
       
= = = =
       
− −
       
a) Find the transition matrix '
B B
P → from '
B to B .
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Q. 3: Determine whether the set of all pairs of real numbers of the form (1, x) with the
operations
( ) ( ) ( ) ( ) ( )
1, 1 ' 1, ' and 1, 1,
y y y y k y ky
+ + = + =
is a vector space. If not then identify the vector space axioms that fail.
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