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**I-Think** Consider a vector space V over a field F. Let K be a subfield of F.

Prove that V over K is a vector space if we retain the same addition operation and restrict scalar multiplication to scalars from K

Work so far

I figure if we just prove that there exists the 0 element, and that addition and scalar multiplication are closed, the statement.

I know there exists a zero element, as 0x=0 for $\displaystyle x\in{V}$ and $\displaystyle 0\in{F}$, and this 0 is also in K

Under scalar multiplication, all scalars from K are also in F, so if it was closed under F it will also be closed under K

But how do I resolve addition if the same operation is retained.