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The origin stays fixed because linearity requires $T(av+bw)=aT(v)+bT(w)$. If we set $v=0$ and $w=0$ (or simply consider the zero vector), $T(0) = T(0\cdot v) = 0\cdot T(v) = 0$.

Conditions: The transformation is linear.; Applying the definition of linearity to the zero vector.

The origin stays fixed because linearity requires $T(av+bw)=aT(v)+bT(w)$. If we set $v=0$ and $w=0$ (or simply consider the zero vector), $T(0) = T(0\cdot v) = 0\cdot T(v) = 0$.

Conditions: The transformation is linear.; Applying the definition of linearity to the zero vector.