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Rewriting a vector as a linear combination of standard basis vectors allows you to use the linearity of the transformation. Instead of computing the full matrix-vector product directly, you can apply the transformation to each basis vector separately (which corresponds to the columns of the matrix) and then combine the results using the original coefficients.

Conditions: The vectors are in $R^3$.; The coefficients are the coordinates of the vector relative to the standard basis.; A is linear: applying it preserves sums and scalar multiples.

Rewriting a vector as a linear combination of standard basis vectors allows you to use the linearity of the transformation. Instead of computing the full matrix-vector product directly, you can apply the transformation to each basis vector separately (which corresponds to the columns of the matrix) and then combine the results using the original coefficients.

Conditions: The vectors are in $R^3$.; The coefficients are the coordinates of the vector relative to the standard basis.; A is linear: applying it preserves sums and scalar multiples.

Subtraction is equivalent to adding the additive inverse. In matrix algebra, multiplying a matrix by the scalar -1 creates its additive inverse.

Conditions: Scalar multiplication by -1 is defined entrywise.; Matrix addition is defined entrywise.; A and B have the same dimensions.

Subtraction is equivalent to adding the additive inverse. In matrix algebra, multiplying a matrix by the scalar -1 creates its additive inverse.

Conditions: Scalar multiplication by -1 is defined entrywise.; Matrix addition is defined entrywise.; A and B have the same dimensions.

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.