In my lecture scripts we introduced the general linear group. We already know that $ L (V, V) = {f: V rightarrow V | f text {is linear} } $ is a ring with unity and therefore $ GL (V) = {x in L (V, V) | x text {is invertible} } $ is a group. I do not understand the proof of the above statement in the script:

I do not understand why $ A_2 ^ {- 1} = A_2 $, there $ A_2 (A_2 (v_1)) = A_2 (-v_1) $

What I do not understand here is the definition of $ A_2 $ we have that $ v_1 neq-v_1 $ and therefore $ A_2 (-v_1) = – v_1 $, On the other hand, a linear map is uniquely determined by the values โโof the basis vectors and thus $ A_2 $ has to be linear and that means $ A_2 (-v_1) = – (A_2 (v_1)) = v_1 $,

But then we have $ v_1 = -v_1 $, This is not the case, for example $ V = mathbb {R} ^ 2 $

Can someone tell me where I am wrong?