Cartesian product of vector spaces

Suppose $V_{1},\ldots,V_{N}$ are vector spaces over a field $\mathbbmss{F}$ . Then the Cartesian product $V_{1}\times\cdots\times V_{N}$ is a vector space when addition and scalar multiplication is defined as follows

	$\displaystyle(u_{1},\ldots,u_{N})+(v_{1},\ldots,v_{N})$	$\displaystyle=$	$\displaystyle(u_{1}+v_{1},\ldots,u_{N}+v_{N}),$
	$\displaystyle k(u_{1},\ldots,u_{N})$	$\displaystyle=$	$\displaystyle(ku_{1},\ldots,ku_{N})$

for $u_{i},v_{i}\in V_{i}$ , $k\in\mathbbmss{F}$ .

For example, the vector space structure of $\mathbbmss{R}^{n}$ if defined as above.

Properties

1.

If $V_{i}$ are vector spaces and $W_{i}\subset V_{i}$ are subspaces, then $W_{1}\times\cdots\times W_{N}$ is a vector subspace of $V_{1}\times\cdots\times V_{N}$ .
2.

The dimension of $V_{1}\times\cdots\times V_{N}$ is $\dim V_{1}+\cdots+\dim V_{N}$ .

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