The inverse of a bijective linear map is linear. If $y_i=T(x_i)$, linearity gives $T(ax_1+bx_2)=ay_1+by_2$, so $T^{-1}(ay_1+by_2)=aT^{-1}(y_1)+bT^{-1}(y_2)$. Bijectivity ensures every target vector has a unique preimage.
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