Suppose $T \in \mathcal{L}(V,W)$, and $v_1,...v_n$ is a base of $V$, is $T(v_1),...T(v_n)$ a base of $W$?
If yes, how to prove it?
The reason I have this question is when I am reading the definition of matrix of a linear map, it said,
Suppose $T \in \mathcal{L}(V,W)$, and $v_1,...v_n$ is a base of $V$, and $w_1,...,w_m$ is a base of $W$. The matrix of $T$ with respect to these bases is the m by n matrix $M(T)$ whose entries $A_{jk}$ are defined by $$Tv_k = A_{1,k}w_1 + ...+ A_{m,k}w_m$$
Based on the definition, for every vector in the base of $V$, after we transformed it by $T$, we will get a vector that can be expressed by $w_1,...,w_m$. And the definition said $w_1,...,w_m$ is a basis. So, I am wondering, if $T(v_1),...T(v_n)$ a base of $W$?
HINT: This is not true.Consider $A:\mathbb{R^n} \to \mathbb{R^n}$ defined by $A(v)= 0$ (by zero i think zero vector).