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Kernel of linear operator

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Kernel of projection.jpg

If $M, L$ are vector spaces and $A: M → L$ is a linear operator, then the kernel of $A$ is the set of elements that $A$ takes to 0: $$\ker A=\{ x\in M:A(x)=0 \}.$$

It's, then, the preimage of $0$ under $f$: $$\ker A=f^{-1}(0).$$

The kernel of the projection of the 3-space to the plane, right, is the z-axis.

Theorem. $$M / \ker A ≌ im A.$$