§ Kernel, cokernel, image

Consider a linear map $T: X \rightarrow Y$. we want to solve for $\{ x : T(x) = y0 \}$.

• If we have an $x0$ such that $T(x0) = y0$, then see that $T(x0 + Ker(T)) = T(x0) + T(Ker(T)) = y0 + 0 = y0$. So the kernel gives us our degrees of freedom: how much can we change around without changing the solution.
• Now consider the cokernel: $Coker(T) = Y / Im(T)$. If we want to find a solution $\{ x : T(x) = y0 \}$. If we have $y0 \neq 0 \in Coker(T)$, then the solution set is empty. The cokernel tells us the obstruction to a solution.