be three vectors in . Prove that these vectors form a basis for .
- Write the vector as a linear combination of .
- Proof. We know that any set of three linearly independent vectors in will span , and thus form a basis. (This is from Theorem 12.10, which is valid for .) Thus, it is sufficient to show that are linearly independent. To that end, let be scalars in , then
From the third equation we have , and so the second equation implies , and finally the third equation implies . Hence, , and are linearly independent
- To express as a linear combination of , let be scalars. Then,
From the third equation we have . Plugging this into the first and second equations we get and . Therefore,