Chapter 8.1, Problem 46P

Let $T:V\to W$ be a linear transformation, and suppose that $\left\{v_1,v_2,\dots ,v_n\right\}$ is a basis of $\ker \left(T\right)$. Prove that every solution to the operator equation,

$T\left(v\right)=w$ (8.1.15)

Is of the form

$v=c_1{v}_1+c_2{v}_2+\cdots +c_n{v}_n+v_p$

Where $v_p$ is the particular solution to equation (8.1.15).