###### tags: `Linear Algebra`
$T:1-1 \implies T^t:$ onto
$T:V \rightarrow W, T^{t}:W^* \rightarrow V^*$

$\forall f \in V^*$ we want to find $g$ such that $T^t(g)=f$
i.e. $g \circ T =f$
so we want $g(T(v))=f(v)$ $\forall v \in V$

so choose a basis $\beta=\{v_1,v_2, \cdots,v_n\}$ of $V$
Since $T$ is $1-1,$ $\{T(v_1),T(v_2), \cdots, T(v_n)\}$ is a linearly independent set in $W$
Then extend $T(v_i)$ to $\{T(v_1),T(v_2), \cdots, T(v_n), w_1, \cdots, w_k\}$ basis of $W$
Then define $g \in W^*$ by
$\ \ \ g(T(v_1))=f(v_1)$
$\ \ \ \ \ \ \ \ \ \ \vdots$
$\ \ \ g(T(v_n))=f(v_n)$
$\ \ \ g(w_i)=0$

所以這樣的話我$V$的basis 是不是可以任取
不一定要先取$f$然後extend成 $\beta^*=\{f,\varphi_1,\cdots,\varphi_n\}$
然後再取 $\beta^*$ 的 dual