I'm currently refreshing my Mathematics knowledge to pass an exam, so I found a list of tasks and got stuck on this one. Could anyone please help me to understand, what am I supposed to even start with?
Here is the task:
A subset $Y$ of a vector space $V$ is called an affine subspace if there exists a $v \in V$ and
there exists a subspace $W$ of $V$ such that:
$$Y = v + W := \{u \in V | \exists w \in W : u = v + w \}$$
Show that:
For all $v' \in Y$ indeed $Y = v´ + W$
For another vector space $Z, f : V \to Z$ is linear, $z \in Z$ and $u \in f^{-1} ({z})$ is valid $f^{-1} ({z}) = u + \ker f$.
For 1. start by showing $Y\subseteq v' + W$. For that, take $u = v + w \in Y$, note that $u = v' + (u - v')$ and check that $u-v'\in W$. The inclusion $Y\supseteq v' + W$ is shown in the same fashion.
For 2. note that any vector of the form $u+v$ with $f(u)=z$ and $f(v)=0$ also satisfies $f(u+v) = z$. Conversely whenever $f(u')=z$ you have $f(u'-u)=z-z=0$ so that $u'-u\in\operatorname{ker}(f)$ and hence $u'=u+(u'-u)\in u+\operatorname{ker}(f)$.