MTH501 assignment no 3 (due date 8 January 2017)
For a nonempty subset of a vector space, under the inherited operations, the following are equivalent statements
 is a subspace of that vector space
 is closed under linear combinations of pairs of vectors: for any vectors and scalars the vector is in
 is closed under linear combinations of any number of vectors: for any vectors and scalars the vector is in .
Idea About Problem no 2
S = {


,


}

Of vectors in the vector space R^{3}, find a basis for span S.
The set S = {v_{1}, v_{2}} of vectors in R^{3} is linearly independent if the only solution of
c_{1}v_{1} + c_{2}v_{2} = 0
is c_{1}, c_{2} = 0.
In this case, the set S forms a basis for span S.
Otherwise (i.e., if a solution with at least some nonzero values exists), S is linearly dependent.
If this is the case, a subset of S can be found that forms a basis for span S.
With our vectors v_{1}, v_{2}, (*) becomes:
c_{1}


+

c_{2}


=


Rearranging the left hand side yields

=


The matrix equation above is equivalent to the following homogeneous system of equations



 the trivial solution only (meaning that S is linearly independent), or
 the trivial solution as well as nontrivial ones (S is linearly dependent).



1 c_{1}


=

0


1 c_{2}

=

0


0

=

0

Therefore the set S = {v_{1}, v_{2}} is linearly independent.
Consequently, the set S forms a basis for span S and many set of solution .