Inner products of vectors

For a real or complex vector space \({V}\), we can generalize another Cartesian structure, the inner product (AKA scalar product, dot product). We define an inner product space as including a mapping from vectors to scalars denoted \({\left\langle v,w\right\rangle }\) (also denoted \({(v,w)}\) or \({v\cdot w}\)). The mapping must satisfy:

  • \({\left\langle v,w\right\rangle =\left\langle w,v\right\rangle ^{*}}\) where \({^{*}}\) denotes complex conjugation
  • \({\left\langle ax+y,bz\right\rangle =a^{*}b\left\langle x,z\right\rangle +b\left\langle y,z\right\rangle }\)
  • \({\left\langle v,v\right\rangle >0}\) except if \({v=0}\), in which case it vanishes

The first requirement implies that \({\left\langle v,v\right\rangle }\) is real, and that the inner product is symmetric for real scalars. The second requirement can be phrased as saying that the inner product is anti-linear in its first argument and linear in its second, or sesquilinear, and the first and second requirements together define a Hermitian form. A real inner product is then bilinear or multilinear, meaning linear in each argument. The third requirement above makes the inner product positive definite.

Δ Sometimes the definitions of both inner product and sesquilinear are reversed to make the second argument anti-linear instead of the first. This is sometimes called the “mathematics” convention, while ours would then be the “physics” convention.

Two vectors are defined to be orthogonal if their inner product vanishes. The orthogonal complement of a subspace \({W}\) of \({V}\) is the subspace of all vectors orthogonal to every vector in \({W}\), i.e. \({W^{\perp}\equiv\left\{ v\in V\mid\forall w\in W\left\langle v,w\right\rangle =0\right\} }\). A basis of \({W}\) together with a basis for its orthogonal complement \({W^{\perp}}\) forms a basis for all of \({V}\).

 

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