CO350 LINEAR PROGRAMMING  SOLUTIONS TO ASSIGNMENT 2
Exercise 1.
Let
{
a
1
,...,a
k
} ⊂
R
m
be a set of linearly independent vectors.
(a) Prove that if
a
k
+1
∈
R
m
is not in the subspace spanned by
a
1
,...,a
k
then the set
{
a
1
,...,a
k
+1
}
is
linearly independent.
(b) Prove that the set
{
a
1
,...,a
k
}
is contained in a basis of
R
m
.
Solution:
(a)
Suppose by contradiction that there exist coefﬁcients
c
1
,...,c
k
+1
∈
R
, not all zero, such that
c
1
a
1
+
c
2
a
2
+
...
+
c
k
+1
a
k
+1
= 0
. As
a
1
,...,a
k
are linearly independent,
c
k
+1
6
= 0
. Then
a
k
+1
=

1
c
k
+1
(
c
1
a
1
+
c
2
a
2
+
...
+
c
k
a
k
)
. Therefore
a
k
+1
is in the subspace spanned by
a
1
,...,a
k
,
contradiction.
(b)
The statement is trivial if
{
a
1
,...,a
k
}
is a basis of
R
m
. If not, then there exists a vector
a
k
+1
∈
R
m
which is not in the subspace spanned by
a
1
,...,a
k
. By part (a), the set
{
a
1
,...,a
k
+1
}
is linearly
independent. We can iterate this process until we have
m
linearly independent vectors (hence a