- Prove that a set
of three vectors in
is linearly independent if and only if its span contains the three unit coordinate vectors.
- State and prove a generalization of part (a) for the space
.
- Proof. Assume
is a basis for
. Then
are in the span of
since they are in
and the span of
is all of
by the definition of basis.
Conversely, assume
are in the span of
. Let
. Since
are in the span of
we know there are scalars
such that
But then if
is any vector we have
Then, if
for three vectors in
then
is a basis (the three vectors must be independent, otherwise there would be two vectors that span
, contradicting Theorem
- Claim. A set
of
vectors in
is a basis for
if and only if its linear span
contains the unit coordinate vectors of
.
Proof. Assumeis a basis for
, then
. Since the unit coordinate vectors of
are in
they are then in
.
Conversely, assume the unit coordinate vectors are in. Then, by the same argument as in part (a) we know
. But, since there are
vectors in
, we know
. Hence,
. Furthermore, the
vectors of
are independent since they span
(if they were dependent, then there would be a set
with fewer than
vectors which would span
, contradicting Theorem