Find a maximal linearly independent subset of each of the following sets of vectors in .

- .
- .
- .

- This set is a dependent set since ; hence, we cannot find a subset of size 4. Then, consider the three element subset . This set is independent since
The third equation gives us ; hence, by the first equation . Therefore, and the set is independent.

- The set is independent since the equation
gives us the equations

The only solution to this system is ; hence, this is a linearly independent subset.

- The set is linearly independent since
gives us the system of equations

Thus, this set is linearly independent.