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 spacewhen 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 $\\dim V_{1}+\\cdots+\\dim V_{N}$ .