- Prove that the vectors
are linearly independent in
.
- Prove that the vectors
are linearly dependent in
.
- Find all
such that the vectors
are linearly dependent in
.
- Proof. We consider the equation
From the first equation we have
and from the third we have
. Plugging these into the second we have
which implies
. Hence,
, so the vectors are independent
- Proof. Consider the equation
The first and third equations implies
. By the second equation we then have
which implies
is arbitrary (since this will hold for all choices of
). Letting
, we have
and we have the equation
Hence, these vectors are linearly dependent
- For
,
,
to be independent we must have
for some nontrivial choice of
. This gives us the equations
If
then
and
can be any value, so the vectors are dependent.
Assume. From the first equation we have
, and from the third we have
. Therefore, the second equation becomes
Thus, we have
.
Therefore, the vectors are dependent if.