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Proofs from Existence, uniqueness and linear operators

All proofs from Existence, uniqueness and linear operators.

[PROOF]

If $y_h\in\ker(T)$ , then

T[y_p+y_h]=T[y_p]+T[y_h]=f+0=f,

so every element of $y_p+\ker(T)$ is a solution. Conversely, if $y$ is any solution of $T[y]=f$ , then

T[y-y_p]=T[y]-T[y_p]=f-f=0.

Thus $y-y_p\in\ker(T)$ , so $y=y_p+y_h$ for some $y_h\in\ker(T)$ .

Part of the thread Kernels and solution sets, in 6 places.

Every solution of $T(x)=b$ is one particular solution plus an element of $\ker T$ .

DFKernel of an operatorthe kernel of a differential operatorCAL 2 / Here