Consider the three vectors in ,

- Determine whether are linearly independent or linearly dependent.
- Find a nonzero vector such that are dependent.
- Find a nonzero vector such that are independent.
- Using the vector obtained in part (c) express the vector as a linear combination of .

- We have
But this implies . Hence, are linearly independent.

- Let . Then, letting , , and we have
Hence, are linearly dependent.

- Let . Then
These equations implies . Hence, are linearly independent.

- So we compute
Since (from the third equation), the second implies . Then the first equation implies . So the third equation implies , and finally the fourth equation implies . Hence,