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**ThePerfectHacker** Let $\displaystyle \bold{x},\bold{y}\in f^{-1}[V]$ and $\displaystyle a\in F$, the base field for the vector spaces. It means by definition that $\displaystyle f(\bold{x}),f(\bold{y}) \in V$. Since $\displaystyle V$ is a subspace it means $\displaystyle f(\bold{x})+f(\bold{y}) \in V$. Therefore, since $\displaystyle f$ is a linear transformation it means $\displaystyle f(\bold{x}+\bold{y}) \in V$ which means $\displaystyle \bold{x}+\bold{y} \in f^{-1}[V]$. Thus, $\displaystyle f^{-1}[V]$ is closed under vector addition. Likewise, $\displaystyle kf(\bold{x}) \in V $ which means $\displaystyle k\bold{x}\in f^{-1}[V]$. Thus, $\displaystyle f^{-1}[V]$ is closed under scalar multiplications. All the other properties for being a vector space are satisfied because $\displaystyle f^{-1}[V]\subseteq X$. Thus, $\displaystyle f^{-1}[V]$ is a vector space over $\displaystyle F$.