Waves and Vectors

[Freeman]

Can a vector be a wave? Or does the wave need to be a number of vectors?

[matt grime]

eh? offer a mathematical defintion of wave and one of a vector and see. don't confuse mathematics with physics, or a model of something with the thing itself either.

[ElijahJones]

There is no mathematical definition of a wave. Waves are physical phenomena. There are functions that gets used to model waves, sine and cosine functions, functions of complex variables, fourier series etc.

 

A vector V in its most general form is simply an ordered n-tuple whose elements are taken from a space S. Since vector operations are usually only meaningful when constructed from the operations of S, we choose S to be a set with some operations that give algebraic structure usually a field. An n-dimensional vector space can be contructed from the set of n-tuples taken over a field F by defining vector addition in natural ways from the operations of F. You must also include F as the set of scalar multipliers satisfying certain conditions. On top of this is usually defined an inner product (the dot product on multivariate calculus). If you want a really hard but excellent text try Dummit and Foote "Absract Algebra". I have 2nd edition but it may be into a third or fourth by now.