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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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    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. $L=0$.