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Corollary: Properties of a Real Scalar Product

(related to Definition: Dot Product, Inner Product, Scalar Product (General Field Case))

Let \(V\) be a vector space over the field of real numbers \(\mathbb R\). Then the scalar product \(\langle \cdot,\cdot\rangle\) on \(V\) fulfills the following properties:

1. linear in the first argument: \(\left\langle \lambda _{1}v+\lambda _{2}w,y\right\rangle =\lambda _{1}\left\langle v,y\right\rangle +\lambda _{2}\left\langle w,y\right\rangle \,\) for all \(\lambda_{1},\lambda _{2}\in \mathbb {R} \), \(v,w,y\in V\),
2. linear in the second argument: \(\left\langle x, \lambda _{1}v+\lambda _{2}w\right\rangle =\lambda _{1}\left\langle x, v\right\rangle +\lambda _{2}\left\langle x, w\right\rangle \,\) for all \(\lambda _{1},\lambda _{2}\in \mathbb {R} \), \(x, v,w\in V\),
3. symmetric: \(\left\langle v,w\right\rangle =\left\langle w,v\right\rangle \,\) for all \(v,w\in V\).
4. symmetric: \(\left\langle v,w\right\rangle =\left\langle w,v\right\rangle \,\) for all \(v,w\in V\).

Table of Contents

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Definitions: 1

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References

Adapted from CC BY-SA 3.0 Sources:

1. Brenner, Prof. Dr. rer. nat., Holger: Various courses at the University of Osnabrück