Let $A\in\text{Mat}_{m,n}(K)$ be an $m\times n$ matrix with coefficients in a field $K$. Then the kernel is often defined in set-builder notation as $\ker(A)=\{x\in K^n|Ax=0 \}$. I would like to know if, like the image of $A$, the kernel can be expressed explicitly in terms of the entries for a general matrix $A=(A)_{ij}$? This seems unlikely as we need to know the zero columns of $A$ after a Gaussian elimination. But, perhaps there is another definition.

Thanks in advance!


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