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**Defunkt** 6=4?

$\displaystyle Ker(T) = \{v\in V : T(v) = 0\}$. You want to find $\displaystyle dim ~ Ker(T)$. By a theorem (it is rather easy to prove), as Bruno had mentioned, we have that $\displaystyle dim ~ V = dim ~ Im(T) + dim ~ Ker(T)$. Since T is onto a space of dimension 4, we get that $\displaystyle dim ~ Im(T) = 4$. However, $\displaystyle dim ~ V = 6$ and so: $\displaystyle dim ~ V = dim ~ Im(T) + dim ~ Ker(T) \Rightarrow 6 = 4 + dim ~ Ker(T) $ $\displaystyle \Rightarrow dim ~ Ker(T) = 6-4 = 2$