$\gamma$ is an inveterible matrix such that $a=b \gamma$, then why is it that we have $\oplus_{i=1}^n Ra_i \subseteq \oplus_{i=1}^n Rb_i$

Question

Let $a=(a_1,a_2,...a_n)$ and $b=(b_1,b_2,...b_n)$ be row vectors with entries in a ring $R$. If $\gamma$ is an invertible matrix such that $a=b \gamma$, then why is it that we have $\oplus_{i=1}^n Ra_i \subseteq \oplus_{i=1}^n Rb_i$

edit: it doesn't need to be a proper subset... Can't find the TeX for that >.<