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$V,W$ two vectorspaces, $\tau: V \rightarrow W$ is a $\mathbb{K}$-linear image.

What is the geometric meaning of $\tau(v) = w$ and $\tau^{-1}(w) = v + Kernel(\tau)$

$v+Kernel(\tau) := ${$v + u: u\in Kernel(\tau)$}

Im confused as to how to think about this theorem. So in "normal" Vectorspaces only the zerovector is mapped onto the zerovector of the image: $\tau(v_0) \rightarrow w_0$. And then of course, the reverseimage of $w = v$:

$\tau^{-1}(w) = v + Kernel(\tau) = v + v_0 = v$

Now first of all, is there an image of a "regular" vectorspace in $\mathbb{R^3}$, so that two vectors are mapped onto the zerovector? (apart form the image that maps everything onto the zerovector)

Below is my interpretation of vectorspaces in 3 dimension and how an image may look like:

enter image description here

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    Actually there are quite a lot of Linear Transformations that map more vectors than just the zero vector to zero. Take a projection, that projects every vector of ℝ onto its first component as an example. – 2017-01-23

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