let be a vector space if we say is a subspace of and we write only if the following is true:
- for its also true that (addition closure)
- for (scalar multiplication closure)
if then is a vector space
let then , let then to be continued..
strictly speaking, a subspace is a vector space included in another larger vector space. therefore, all properties of a vector space, such as being closed under addition and scalar multiplication still hold true when applied to the subspace.
let the set of vectors , consider the following set: by the definition of this set, a vector is in the set only if for to be a subspace of , it has to meet 3 conditions: condition 1: , in this case condition 2: addition closure, meaning if then
condition 3: multiplication closure, meaning if then