Answer
See below
Work Step by Step
Let $T:V \rightarrow W$ be a linear transformation
Suppose $w \in Rng(T)$. There exists $v \in V$ such as $T(v)=w$
then $[w]_C=[T(v)]_C$
From Theorem 6.4.5, we have $[T(v)]_C=[T]^C_B[v]_B\\
\rightarrow [w]_C=[T]^C_B[v]_B$
Hence, $[w]_C$ is a column of the matrix $[T]^C_B$
For the second part, suppose $[w]_C \in$ colspace $([T]^C_B)$
Thus, $[w]_C=[T]^C_B[v]_B$
Since $[T(v)]_C=[T]^C_B[v]_B \rightarrow [w]_C=[T(v)]_C \rightarrow w=T(v)\\
\rightarrow w\in Rng(T)$
