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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 $\neq 2$)2012-04-30
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    @DavideGiraudo: What do you mean with characteristic? So we have 2 linear transformations that are not equal with the same Kernel, right?2012-04-30
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    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 $0$.2012-04-30
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    @DavideGiraudo: We are working with real numbers, the solution system has the <(0,0,0)> solution as one and only.2012-04-30
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    The solutions of $l(x)=0$ are the same as $2l(x)=0$.2012-04-30
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    @DavideGiraudo: Thank you!2012-04-30
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    They *could* be identical, but they don't have to.2012-04-30

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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. $L=0$.