Relation between isomorphism and bases

Question

Let $V$ and $W$ be finite dimensional vector spaces over $\Bbb{F}$ and $T : V → W$ is a linear map. Prove that $T$ is an isomorphism if and only if ${T(v_1 ),T(v_2 ),...,T(v_n)}$ is a basis for $W$ whenever ${v_1 ,v_2 ,...,v_n}$ is a basis for $V$.

Answer 1

Let $T:V \rightarrow W$ be an isomorphism then $Ker(T)=\{0\}$ and $v_1,...,v_n$ be a basis

$a_1T(v_1)+...+a_nT(v_n)=0$ $\Rightarrow$ $T(a_1v_1+...+a_nv_n)=0$

$T(a_1v_1+...+a_nv_n)=0$ $\Rightarrow$ $a_1v_1+...+a_nv_n=0$

$a_1v_1+...+a_nv_n=0$ $\Rightarrow$ $a_1=...=a_n=0$

which means $T(v_1),...,T(v_n)$ forms a basis

Let ${T(v_1 ),T(v_2 ),...,T(v_n)}$ be a basis and $\{v_1,...,v_n\}$ be a basis for $V$

$T(a_1v_1+...+a_nv_n)=T(b_1v_1+...+b_nv_n)=0$ $\Rightarrow$ $T((a_1-b_1)v_1+...+(a_n-b_n)v_n)=0$

$T((a_1-b_1)v_1+...+(a_n-b_n)v_n)=(a_1-b_1)T(v_1)+...+(a_n-b_n)T(v_n)=0$ $\Rightarrow$ $a_1=b_1 ,...,a_n=b_n$

which means all elements of $Ker(T)$ are equal each other so $Ker(T)=\{0\}$ that implies $T$ is an isomorphism