Let , where and are the unit coordinate vectors in .
- Prove that is linearly independent.
- Prove that is in the span of .
- Express as a linear combination of the vectors in .
- Prove that .
- Proof. We have
But then, implies . Hence,
- Proof. Let and . Then,
- We have . Then,
Thus, and is a solution.
- Proof. Since and are two linearly independent vectors in we have by Theorem 12.10 (page 466 of Apostol) that