# Prove that If W is a subspace of a vector space

Prove that
If W is a subspace of a vector space V and ${w}_{1},{w}_{2},\dots ,{w}_{n}$ are in W, then ${a}_{1}{w}_{1}+{a}_{2}{w}_{2}+\dots +{a}_{n}{w}_{n}\in W$ for any scalars ${a}_{1},{a}_{2},\dots ,{a}_{n}$.
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Linda Birchfield
Here W is a subspace of a vector space V and ${w}_{1},{w}_{2},\dots ,{w}_{n}$ are in W.
From above, ${w}_{1}\in W,{w}_{2}\in W,\dots .,{w}_{n}\in W$.
Since, W is closed under scalar multiplication ${a}_{1}{w}_{1}\in W,{a}_{1}{w}_{2}\in W,\dots .,{a}_{1}{w}_{n}\in W$ for any scalars ${a}_{1},{a}_{2},\dots ,{a}_{n}$.
Hence, ${a}_{1}{w}_{1}+{a}_{2}{w}_{2}+\dots +{a}_{n}{w}_{n}\in W$ for any scalars ${a}_{1},{a}_{2},\dots ,{a}_{n}$.