# Orthogonality of vectors

Geometrically, one can alternatively take the approach that “orthogonality” is the fundamental structure to be added to a real vector space $${V=\mathbb{R}^{n}}$$. Orthogonality is defined by requiring that:

1. Any subspace $${W}$$ defines an orthogonal complement $${W^{\bot}}$$ such that only the zero vector is contained in both spaces (an orthogonal decomposition)
2. If $${v}$$ is orthogonal to $${w}$$, then $${w}$$ is orthogonal to $${v}$$

One can then look for bilinear forms that vanish for orthogonal vector arguments. A degenerate form features at least one vector $${v}$$ that is orthogonal to every other vector in $${V}$$, thus violating (1) by being in both $${V}$$ and $${V^{\bot}}$$. By linearity, (2) can only be satisfied if the form is either symmetric or anti-symmetric. Thus our only two candidates are a pseudo inner product or a nondegenerate 2-form. The latter is also called a symplectic form, and can only exist in even-dimensional vector spaces.