If $T:V\to W$ is a bijective linear map, its inverse is linear. Write $y_i=T(x_i)$; then $T^{-1}(ay_1+by_2)=T^{-1}(T(ax_1+bx_2))=ax_1+bx_2$. The inverse is already bijective, so it is an isomorphism.
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