# orthonormal basis, basis of subspace

#### ricta02

##### New member
im completely stuck I know an orthogonal vector is one whose dot product is 0 but don't know how to go about this problem

#### tkhunny

##### Moderator
Staff member
How do you prove "Perpendicular"?

Have you considered an "Orthoginalization" process?

#### ricta02

##### New member
a vector is perpendicular to another vector is 9a-9b-1c-d=0
the gram schmitd using only one vector gives me only one vector and its orthonormal I just need orthogonal

#### daon2

##### Full Member
You want the basis of the nullspace of [9,-9,-1,-1]~[1,-1,-1/9,-1/9]. The nullspace is orthogonal to the row space.

The solution set is almost immediate:

$$\displaystyle \begin{pmatrix}x+y/9+z/9\\x\\y\\z\end{pmatrix} = x\begin{pmatrix}1\\1\\0\\0\end{pmatrix}+y\begin{pmatrix}1/9\\0\\1\\0\end{pmatrix}+z\begin{pmatrix}1/9\\0\\0\\1\end{pmatrix}$$

You can remove the fractions if desired by considering an appropriate multiple.