Cartesian product of vector spaces

Suppose V1,,VN are vector spacesMathworldPlanetmath over a field 𝔽. Then the Cartesian product V1××VN is a vector space when addition and scalar multiplication is defined as follows

(u1,,uN)+(v1,,vN) = (u1+v1,,uN+vN),
k(u1,,uN) = (ku1,,kuN)

for ui,viVi, k𝔽.

For example, the vector space structure of n if defined as above.


  1. 1.

    If Vi are vector spaces and WiVi are subspacesPlanetmathPlanetmath, then W1××WN is a vector subspace of V1××VN.

  2. 2.

    The dimensionPlanetmathPlanetmath of V1××VN is dimV1++dimVN.

Title Cartesian product of vector spaces
Canonical name CartesianProductOfVectorSpaces
Date of creation 2013-03-22 15:16:06
Last modified on 2013-03-22 15:16:06
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 8
Author Mathprof (13753)
Entry type Definition
Classification msc 16-00
Classification msc 13-00
Classification msc 20-00
Classification msc 15-00