Editing Orthogonal complement (section)

==Example==

Let <math>V = (\R^5, \langle \cdot, \cdot \rangle)</math> be the vector space equipped with the usual [[dot product]] <math>\langle \cdot, \cdot \rangle</math> (thus making it an [[inner product space]]), and let <math display="block">W = \{\mathbf{u} \in V: \mathbf{A}x = \mathbf{u},\ x\in \R^2\},</math> with
<math display="block">\mathbf{A} = \begin{pmatrix}
1 & 0\\
0 & 1\\
2 & 6\\
3 & 9\\
5 & 3\\
\end{pmatrix}.</math>
then its orthogonal complement <math display="block">W^\perp = \{\mathbf{v}\in V:\langle \mathbf{u},\mathbf{v}\rangle = 0 \ \ \forall \ \mathbf{u} \in W\}</math> can also be defined as <math display="block">W^\perp = \{\mathbf{v} \in V: \mathbf{\tilde{A}}y = \mathbf{v},\ y \in \R^3\},</math> being
<math display="block">\mathbf{\tilde{A}} =
\begin{pmatrix}
-2 & -3 & -5 \\
-6 & -9 & -3 \\
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 
\end{pmatrix}.</math>

The fact that every column vector in <math>\mathbf{A}</math> is orthogonal to every column vector in <math>\mathbf{\tilde{A}}</math> can be checked by direct computation. The fact that the spans of these vectors are orthogonal then follows by bilinearity of the dot product. Finally, the fact that these spaces are orthogonal complements follows from the dimension relationships given below.