# Can the kernel of a general matrix $A \in\text{Mat}_{m,n}(K)$ be expressed in terms of its entries?

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.