Let $T : V → W$ be a linear transformation. Let $w ∈ W$ and let $u_0 ∈ V$ satisfy $$T(u_0) = w$$
Show that $u ∈ V$ is a solution of the equation $T(u) = w$ if and only if $u = u_0 + v$, where $T(v) = 0$
The proof of this consists of two parts:
1)Prove that $T(u)=T(u_0+v)=T(u_0)+T(v)=w+0=w$
2)Prove that all $u$ that satisfy $T(u)=w$ are in the form $u=u_0+v$
How should I go about proving the second part?
Hint: To show $2)$, you need to show that if $u\in V$ is such that $T(u) = w$, then we must have $u = u_0 + \color{magenta}{v}$ for some $v\in V$ with $T(v) = 0$. It thus suffices to show that $$\color{blue}{T(u-u_0) = 0}$$ (since $u = u_0 + \color{magenta}{(u-u_0)}$). Can you show the last equality?