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.

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