Condition on which this equation holds It is well known that for a linear transformation on an Euclidean space $V$, whose matrix under a certain basis we denote by $A$, we have 
$$Ker(A)\oplus R(A)=V$$
in which $R$ denotes the row space. 
Also, since the column space (denoted by $C(A)$) is exactly $Im(A)$, we have
$$dim(Ker(A))+dim(C(A))=dimV$$
However, this is NOT sufficient to say
$$Ker(A)\oplus C(A)=V$$
is also true.  And I think I have found one counterexample. 
But I'm still curious. Because I find out that at least for one trivial case where $Ker(A)$ is null space, this equation holds. So maybe there are some more cases where this equation, although not generally true, will also hold? Or maybe trivial cases are the only possibility where it holds? I'm sorry I'm not good at finding counterexamples. So I can neither contradict or prove my guesses.  And I need some help here. 
Best regards!

In general, the equality (direct sum) is not always true. However, you can do it for projection. If A satisfies $A^{2}=A$, then $V=KerA \bigoplus ImA $, a typical example of those is a diagonal $n \times n$ matrix, with 0 and 1 as eigenvalues!
