Cartesian product of vector spaces
Suppose ${V}_{1},\mathrm{\dots},{V}_{N}$ are vector spaces^{} over a field $\mathbb{F}$. Then the Cartesian product ${V}_{1}\times \mathrm{\cdots}\times {V}_{N}$ is a vector space when addition and scalar multiplication is defined as follows
$({u}_{1},\mathrm{\dots},{u}_{N})+({v}_{1},\mathrm{\dots},{v}_{N})$  $=$  $({u}_{1}+{v}_{1},\mathrm{\dots},{u}_{N}+{v}_{N}),$  
$k({u}_{1},\mathrm{\dots},{u}_{N})$  $=$  $(k{u}_{1},\mathrm{\dots},k{u}_{N})$ 
for ${u}_{i},{v}_{i}\in {V}_{i}$, $k\in \mathbb{F}$.
For example, the vector space structure of ${\mathbb{R}}^{n}$ if defined as above.
Properties

1.
If ${V}_{i}$ are vector spaces and ${W}_{i}\subset {V}_{i}$ are subspaces^{}, then ${W}_{1}\times \mathrm{\cdots}\times {W}_{N}$ is a vector subspace of ${V}_{1}\times \mathrm{\cdots}\times {V}_{N}$.

2.
The dimension^{} of ${V}_{1}\times \mathrm{\cdots}\times {V}_{N}$ is $dim{V}_{1}+\mathrm{\cdots}+dim{V}_{N}$.
Title  Cartesian product of vector spaces 

Canonical name  CartesianProductOfVectorSpaces 
Date of creation  20130322 15:16:06 
Last modified on  20130322 15:16:06 
Owner  Mathprof (13753) 
Last modified by  Mathprof (13753) 
Numerical id  8 
Author  Mathprof (13753) 
Entry type  Definition 
Classification  msc 1600 
Classification  msc 1300 
Classification  msc 2000 
Classification  msc 1500 