- Let
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,