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Matrix addition is performed by adding corresponding entries. If two matrices have the same number of rows and columns, their sum is a new matrix where each element $(i,j)$ equals the sum of the elements at position $(i,j)$ in both original matrices.

Conditions: Both matrices must have identical dimensions (same $m \times n$).

Matrix addition is performed by adding corresponding entries. If two matrices have the same number of rows and columns, their sum is a new matrix where each element $(i,j)$ equals the sum of the elements at position $(i,j)$ in both original matrices.

Conditions: Both matrices must have identical dimensions (same $m \times n$).

Multiplying a 3x3 matrix by a vector $(x, y, z)$ scales each column of the matrix (which represents a transformed basis vector) by the corresponding input coordinate and sums the results. This works because linear transformations preserve addition and scalar multiplication.

Conditions: Matrix is 3x3 representing a linear transformation; Input vector has coordinates $(x, y, z)$; Transformation is linear

Multiplying a 3x3 matrix by a vector $(x, y, z)$ scales each column of the matrix (which represents a transformed basis vector) by the corresponding input coordinate and sums the results. This works because linear transformations preserve addition and scalar multiplication.

Conditions: Matrix is 3x3 representing a linear transformation; Input vector has coordinates $(x, y, z)$; Transformation is linear