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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.

Matrix addition requires matching dimensions so that every entry in one matrix has exactly one corresponding partner in the other matrix. Without identical row and column counts, some positions would lack a pair to add, making the operation undefined under standard rules.

Conditions: Standard definition of matrix addition applies.; Matrices represent rectangular arrays of numbers.

The determinant represents the signed area scaling factor of the linear transformation defined by the matrix. Geometrically, the columns of the matrix form two vectors that span a parallelogram.

Conditions: A is $a 2\times 2$ matrix.; The columns of A are interpreted as two vectors in the plane.; Use ordinary Euclidean area in standard orthonormal coordinates.

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.

Standard matrix addition and subtraction are defined for matching dimensions, because every position requires a corresponding entry. Different dimensions leave unmatched positions under this standard rule; a separately specified new convention would be a different operation.

Conditions: Matrices have different dimensions (e.g., one is 3x2 and the other is 2x2).

When computing the composition $A \circ B$, the matrix B represents the first transformation. Its columns are the images of the standard basis vectors under B.

Conditions: A and B are 3x3 matrices representing linear transformations.; The goal is to find the matrix representation of the composite transformation.