# 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. 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. 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 CartesianProductOfVectorSpaces 2013-03-22 15:16:06 2013-03-22 15:16:06 Mathprof (13753) Mathprof (13753) 8 Mathprof (13753) Definition msc 16-00 msc 13-00 msc 20-00 msc 15-00