A given set of vectors
is said to be linearly independent
there is no nontrivial representation
of 
in terms of V. If a set
is not linearly independent, it is
linearly dependent .
For simplicity we will refer to
a linearly independent set as independent
and a linearly dependent set as dependent.
Examples
,
.
So has the nontrivial representation
in terms of V
V is linearly dependent.
,
= (0,0,0)
.
Thus the only representation of in terms
of V is the trivial one
V is linearly
independent.
Proof
Assume, to the contrary, that
is dependent.
Then there are vectors
(as well as scalars
such that
),
contradicting the property that has no members!
Thus is independent.
Proof
Let 
be given with 
.
[Show 
is independent.]
Assume to the contrary, that
is dependent
for some 
(since ),
a contradiction!
Thus is independent.
Proof
Using a combining coefficient of 1 with
and combining coefficients of zero
with the remaining vectors of V produces a nontrivial representation
of V is dependent.
Proof
the only representation of is the trivial representation.
Thus
is independent.
Proof
Let 
be given.
Assume V is dependent.
Then there are
, not all zero,
such that
.
Let j be the largest integer such that 
.
[Show j > 1.] Assume, to the contrary, that j = 1
(since 
), contradicting 
!
Thus j > 1
,
a linear combination of the preceeding vectors.
Assume some vector 
with j > 1
is a linear combination of
the preceeding vectors
.
Then there are some 
such that
V is dependent.
Proof Left as an exercise.
Proof Left as an exercise.
Proof Left as an exercise.
Proof Left as an exercise.
Proof Left as an exercise.
Proof Left as an exercise.
Proof
Let
be given.
Assume A is orthogonal.
For each i
such that 
,
.
Suppose
.
Then
for each i
such that ,
(since 
)
A is independent.
Proof
Let
be a given orthogonal set with
.
Assume
.
By Prop. 
,
is an orthogonal set, since 
.
By Prop. ,
is independent.
