If $U$ is a subspace of the codomain of a linear map $T$, then $T^{-1}(U)$ is a subspace of its domain. It contains zero, and if $T(x),T(y)\in U$, then $T(ax+by)=aT(x)+bT(y)\in U$. This applies to inverse images even when $T$ has no inverse map.
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