Orthogonality and Orthonormality

Linear algebra has a great capability for making geometric statements, due to the inherent visual structure in many spaces like \(\mathbb{R}^n\). Line segments in such spaces are normally known as orthogonal, or perpendicular, if they intersect at an angle of \(90^\circ\) and orthonormal if they additionally each have a length of \(1\). In the same way, vectors are known as orthogonal if they have a dot product (or, more generally, an inner product) of \(0\) and orthonormal if they have a norm of \(1\). It turns out these two definitions are the same, and the connection between linear algebra and geometry quite strong.

Orthogonality (and orthonormality) is necessary to project vectors onto subspaces, find better estimates of nonlinear objects, and measure many properties of vector spaces.

The length of a vector \(v\) is \(\sqrt{\langle v, \, v \rangle} = \sqrt{v \cdot v}\).

Two vectors \(v\) and \(w\) are said to be orthogonal if \(v \cdot w = 0\). They are orthonormal if each vector has length \(1\).

Orthogonality and orthonormality allow for a much stronger geometric understanding of direction in vector spaces.

• Dot Products
• Gram-Schmidt Process
• Linear Algebra