be vectors in .
- Prove that these three vectors are independent.
- Express the unit coordinate vectors and as linear combinations of these three vectors.
- Express the vector as a linear combination of .
- Prove that this set is a basis for .
- Proof. The equation implies
But this implies . Hence, are linearly independent
- First, we have
From these we have , and . Thus, .
Next, we have
This implies , and . Therefore, .
- We compute,
Therefore, , and . Hence,
- Proof. By Theorem 12.10, any set of 3 linearly independent (part (a)) vectors in is a basis for