- 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 becomesThus, we have .
Therefore, the vectors are dependent if .