When does the kernel of a function equal the image?
Thanks in advance
I think you mean a linear transformation, not a function. They're not the same thing.
I haven't thought about this question, but I have a notion. By the rank-nullity theorem, the rank would have to be zero in that case so the nullity would be n (where n is the number of columns). Any pivot column would mean a rank greater than 0. So there must be no pivot columns. The only way I can picture that is a matrix with all zeroes.
There are several meanings for kernel of a function,
The OP should have defined terms.
Post Script: Here is a second reference.
If this had said "null space", rather than "kernel", then I would say that a vector space, and so "linear transformation", but "kernel" is more general.
The "image" of a function, f, is $\displaystyle \{y | y= f(x)$ for some x}, the "kernel" is $\displaystyle \{ x| f(x)= 0\}$. If the kernel is equal to the image, then, for all x, f(x) is i the image and so f(f(x))= 0 and, conversely, if f(y)= 0, there exist x such that f(x)= y.