If a Vector Space $V$ has dimension $n$ and $T:V\to V$ is linear, then is the following true? Suppose we have that a basis for $V$ is 
$B=\{v_1,v_2,\ldots ,v_n\}$ and $v_1,v_2,\ldots,v_k$ are a basis for $\operatorname{im}(T)$.
Does it follow that $v_{k+1}, v_{k+2},\ldots, v_n$ is a basis for $\ker(T)$? What if $T$ is idempotent? 

Suppose $T(v_1) = v_1, \quad T(v_2) = v_2, \quad T(v_3) = v_1+v_2.$ Then $\{v_1,v_2\}$ is a basis of $\operatorname{im}(T)$ but $\{v_3\}$ is not a basis of $\ker(T).$ In this case $\{v_1+v_2-v_3\}$ is a basis of $\ker(T).$
