General question in linear algebra:
Let $T:V\to W$ be a linear transformation and $\{v_1\,\dots,v_n \}$ a basis for B. If $\{v_1\,\dots,v_k \}$ is a basis for $kerT$.
Can I conclude that the vectors $\{T(v_{k+1}), \dots , T(v_n)\}$ are linearly independent in W?
A clear explanation would be appreciated.
Yes, you can indeed conclude that those vectors are linearly independent.
In particular, let $c_{k+1},\dots,c_n$ be such that $$ c_{k+1}T(v_{k+1}) + \cdots + c_n T(v_n) = 0 $$ we can then write $$ T(c_{k+1}v_{k+1} + \cdots + c_n v_n) = 0 $$ which is to say that $c_{k+1}v_{k+1} + \cdots + c_n v_n \in \ker T$, which is the span of $\{v_1,\dots,v_k\}$. However, since the set $\{v_1,\dots,v_n\}$ is linearly independent, the vector $c_{k+1}v_{k+1} + \cdots + c_n v_n$ can only be in the span of $\{v_1,\dots,v_k\}$ if $c_{k+1} = \cdots = c_n = 0$.