Let $F$ a field, $V$ and $W$ vector spaces, $T$ a linear transformation from $V$ to $W$, if $w$ belongs $W$ and $v$ in $V$ such that $Tv = w$, if $x$ belongs $V$ then $Tx = w$ if and only if $x = v + k$ with $k$ in $\operatorname{ker}(T)$.
I can't solve this, someone can help me please.
We are given that $T:V\to W$ is linear with $T(v)=w$.
First, suppose that $T(x)=w$. Then $$ T(x-v)=T(x)-T(v)=w-w=\mathbf 0 $$ so that $x-v\in\ker T$. That is, there exists a $k\in\ker T$ such that $x-v=k$. Hence $x=v+k$ as required.
Conversely, suppose that $x=v+k$ where $k\in\ker T$. Then $$ T(x)=T(v+k) $$ Can you manipulate this equation to conclude that $T(x)=w$?