Product space/Vector space/Explicit/Exercise

Show that the product space ${}K^{n}$, for a field ${}K$, is, with componentwise addition and scalar multiplication, the properties

1. ${}r(su)=(rs)u$,
2. ${}r(u+v)=ru+rv$,
3. ${}(r+s)u=ru+su$,

4. ${}1u=u\,,$

hold.

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