This document contains 7 questions for a linear algebra group assignment. The questions cover topics such as determining if a polynomial can be expressed as a linear combination of other polynomials; properties of subspaces and spans of subsets of a vector space; determining if a set is linearly dependent or independent; evaluating functions and determining if they represent linear mappings; solving a system of linear equations; and classifying vectors as belonging to the range of a function. Students are to answer one question each in groups of 8 and submit their work by a specified deadline.

1. MATHEMATICS AND PHYSICAL SCIENCE DEPARTMENT
MATH 1206-LINEAR ALGEBRA I GROUP WORK
ANSWER ALL QUESTIONS
Instructions
i. To be done in groups of 8 students (each one question)
ii. To be submitted on or before 21/09/2020
1) Determine whether the polynomial 3 2
4 2 6x x  can be expressed as a linear combination
of 3 2
2 4 1x x x   and 2 2
3 6 4x x x   .
2) Let 1S and 2S be subsets of a vector space V. Prove that
1 2 1 2span(S S ) span(S ) span(S )   . Give an example in which 1 2span(S S ) and
1 2span(S ) span(S ) are equal and one in which they are unequal.
3) Show that if A is an m n matrix, then the null A is a subspace of n
4) Determine whether the following sets are linearly dependent or linearly independent
i.
1 3 2 3 7 4 2 3 11
, ,
4 0 5 6 2 7 1 3 2
S
         
       
          
ii.  (1, 1,2),(2,0,1),( 1,2, 1)W    
5) Suppose that 2 3
:f  and 3 2
g :  are functions defined by
1 2 1 2 1 2( , ) ( , , )f x x x x x x  and 22 31 3 1( , , ) ,3 )(xx x xg x x  , find g f and deduce whether
it is a linear mapping.
6) Let 3 3
:f  be defined by 1 2 3 1 2 2 3 1 2(x ,x ,x ) = (x + x ,x -2x ,x +2x )f , for
3
(1,3, 2)v    determine whether ( )v R f
7) find the dimension and a basis for the general solution W of the following system of linear
equation
1 2 3 4
1 2 3 4
1 2 3 4
2 4 5 3 0
3 6 7 4 0
5 10 11 6 0
x x x x
x x x x
x x x x
   
   
   