I have a problem that says: "By using the Gram Schmidt process(if you need it), find an orthonormal basis B in the subspace: $A=\{ u \in R^3 \ | \ x_1-4x_2-x_3=0 \}$
I chose as the basis, $B=\{[1 \ -4 \ -1]\}$ and normed it by dividing each entry with $\sqrt{16}$. Is it correct? How could I use Gram Schmidt? I know how to use it when I have 2+ basis vectors.
Best Answer
Your basis is incorrect. The subspace $A$ has dimension $2$, not $1$, so you need to start with two basis vectors, not one.
In addition, your vector $B$ is not even in the subspace: if you plug in $x_1=1$, $x_2=-4$, $x_3=-1$ into $x_1-4x_2-x_3$, you do not get $0$.
So, first write out the solutions to $x_1-4x_2-x_3=0$, e.g., in parametric form. This will give you the two basis vectors. Then apply Gram-Schmidt to the basis you got.
(In the special case where the vector space is $1$-dimensional, the orthogonalization part of Gram-Schmidt is empty, so all you need to do is normalize a single basis vector).