# image & kernel

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• March 21st 2013, 11:10 AM
jojo7777777
image & kernel
When does the kernel of a function equal the image?
Thanks in advance
• March 21st 2013, 02:15 PM
zhandele
Re: image & kernel
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.
• March 21st 2013, 02:23 PM
Plato
Re: image & kernel
Quote:

Originally Posted by zhandele
I think you mean a linear transformation, not a function. They're not the same thing.

There are several meanings for kernel of a function,

The OP should have defined terms.

Post Script: Here is a second reference.
• March 21st 2013, 04:07 PM
HallsofIvy
Re: image & kernel
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 $\{y | y= f(x)$ for some x}, the "kernel" is $\{ x| f(x)= 0\}$. 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.
• March 21st 2013, 04:08 PM
Ant
Re: image & kernel
Quote:

Originally Posted by jojo7777777
When does the kernel of a function equal the image?
Thanks in advance

From my understanding of the terms; this could only happen if the function was the 0 map (the map which maps all elements in the domain to 0) from (say) X into X.
• March 23rd 2013, 01:02 AM
jojo7777777
Re: image & kernel
In my question I refer to linear transformation , and I meant to ask whether there are some features in the image that can tell me about the fact that the null-space and image are equal?