Let
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 haveThis 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