# Identical linear transformations when their kernel is the same?

Given this subspace $\langle (0,0,0)\rangle$ as a solution set of a homogeneous system of linear equations, so it is a Kernel of a linear transformation.

If two linear transformations have the same Kernel, could they be identical?

Thank you!

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$l$ and $2l$ have the same kernel (if you work in characteristic $\neq 2$) –  Davide Giraudo Apr 30 '12 at 16:07
@DavideGiraudo: What do you mean with characteristic? So we have 2 linear transformations that are not equal with the same Kernel, right? –  Chris Apr 30 '12 at 16:10
I should add "field of characteristic $2$", i.e. such that $1+1=0$. If you work with the field of real or complex numbers, and $l$ is a linear map, then $l$ and $2l$ have the same kernel, but are not identical, except if $l$ is identically $0$. –  Davide Giraudo Apr 30 '12 at 16:12
@DavideGiraudo: We are working with real numbers, the solution system has the <(0,0,0)> solution as one and only. –  Chris Apr 30 '12 at 16:27
They could be identical, but they don't have to. –  Arturo Magidin Apr 30 '12 at 17:11

Of course, if two linear transformation are equal, then they have the same kernel. But if two linear transformations have the same kernel, we are not sure that they are equal. For example, if $L$ is a linear transformation, so is $2L$, and $2L$ and $L$ have the same kernel. They are equal if and only if $2L=L$ i.e. $L=0$.