Isomorphism and Basis

Question

I started reading Linear algebra done wrong and am confused at a theorem that he leaves the proof off to the reader. The Theorem is if A: V -> W is an isomorphism, and v1, v2, ... vn are a basis in V. Then the system Av1, ..., Avn is a basis in W. I had tried a proof and attached it below but it is wrong I am almost certain. I have made no use of the face that there is an Isomorphism between the Spaces where does that come in and why is my proof not complete and or correct?

Answer 1

For any $w\in W$, there exists $v\in V$ such that $Av=w$ because A is surjective.

Since $v_1,...v_n$ form a basis for $V$, $v=a_1v_1+...+a_nv_n$ for some $a_i$ and $Av=A(a_1v_1+...+a_nv_n)=a_1Av_1+...a_nAv_n=w$. Therefore, the $Av_i$ span $W$.

Suppose $b_1Av_1 +...+b_nAv_n=0$.

$b_1Av_1 +...+b_nAv_n=A(b_1v_1+...+b_nv_n)$ by linearity. Since $A$ is an isomorphism it is injective so $b_1v_1+...+b_nv_n=0$.

Therefore, $b_1v_1+...+b_nv_n=0\implies b_i = 0$ for all $i$ because the $v_i$s form a basis and so are linearly independent. Since the $b_i$s are all $0$, $Av_1,...,Av_n$ are linearly independent and therefore, a basis.

Answer 2

I can't say John's solution is wrong. However, Treil's book is rigorous. By this point, he hasn't mentioned surjective at all. I think his intention is to use the definition. Here is my take.