I was thinking about kernel and image and wondered: If $T$ and $S$ are linear transformations from $V$ to $V$, when do we have ker $T$ = im $S$ and im $T$ = ker $S$? Is there any significance to this? Or is it just something trivial? I know it is possible when $T = S$, but how about when $T \neq S$?
I was drawing pictures and it seems that if this is possible, then $T$ and $S$ would sort of be “dual” to each other, which I’ve tried to illustrate in my drawing below. (I’m using “dual” in a general, colloquial sense, not in the sense of dual map.) They map into each other’s kernels. Another simple observation is that $T \circ S = S \circ T = 0$.
Edit: an answer below has mentioned that if $P$ is any projection, then $T = P$ and $S = I – P$ work. Some more questions I have are:
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Are these the only possibilities?
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Observe that in this case (if we have an inner product) each map's image is orthogonal to its kernel, and we can write $V$ as a direct sum of these two subspaces. Is this a necessary condition in general?
Best Answer
An example where this happens is when $T=P$ and $S=I-P$ where $P$ is any projection.