Let $f:V'\rightarrow V$ be a linear function and $A$ be an affine subspace in V, then $f^{-1}(A)$ is an affine subspace in $V'$

Question

The fiber of a set is defined as: $f^{-1}(A) = \{x\in V' : f(x) \in A \}$ and the empty set is neither considered as a linear subspace nor an affine subspace.

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Let $A=a+U$ with $a \in V$ and $U$ being a linear subspace of $V$.

I have already proven that $f^{-1}(U)$ is a linear subspace of $V'$, which I'm sure will come in handy.

I think the idea is that $a$ is being transformed to a new $a'$ whereas $U$ will be transformed to a new subspace $f^{-1}(U)$, but I can't fill in the gaps.

$ f^{-1}(A)=f^{-1}(a+U) \,\longrightarrow\, a'+f^{-1}(U) $

Help would be really appreciated!

Answer 1

If $f^{-1}(A)$ is non-empty, choose some $a'$ in it. Then, $f(a')\in a+U$, or equivalently: $$a+U=f(a')+U.$$ Therefore, $$x\in f^{-1}(A)\iff f(x)\in f(a')+U\iff f(x-a')\in U\iff x-a'\in f^{-1}(U),$$ so that $$f^{-1}(A)=a'+f^{-1}(U).$$