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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

Tracking the images of the standard basis vectors determines the destination of every other point because any arbitrary vector can be expressed as a linear combination of these basis vectors. Since a linear transformation preserves vector addition and scalar multiplication, the transformation of the arbitrary vector is exactly the same linear combination applied to the transformed basis vectors.

Conditions: The transformation is linear.; Working in a 2D plane with standard basis vectors $\hat{i} = [1, 0]^T$ and $\hat{j} = [0, 1]^T$.

Tracking the images of the standard basis vectors determines the destination of every other point because any arbitrary vector can be expressed as a linear combination of these basis vectors. Since a linear transformation preserves vector addition and scalar multiplication, the transformation of the arbitrary vector is exactly the same linear combination applied to the transformed basis vectors.

Conditions: The transformation is linear.; Working in a 2D plane with standard basis vectors $\hat{i} = [1, 0]^T$ and $\hat{j} = [0, 1]^T$.

To compute the matrix for the composition $A \circ B$, apply the transformation $A$ to each column of the matrix $B$. The resulting vectors form the columns of the matrix $A \circ B$.

Conditions: A and B are both 3x3 matrices.; The operation is performed over the real numbers.; These are real linear maps with compatible input and output spaces, represented using standard column-vector coordinates.

To compute the matrix for the composition $A \circ B$, apply the transformation $A$ to each column of the matrix $B$. The resulting vectors form the columns of the matrix $A \circ B$.

Conditions: A and B are both 3x3 matrices.; The operation is performed over the real numbers.; These are real linear maps with compatible input and output spaces, represented using standard column-vector coordinates.

Matrix-vector multiplication in 3D works by scaling the columns of the matrix by the corresponding coordinates of the input vector and summing the results. The coordinates $(x, y, z)$ act as scalar multipliers for the transformed basis vectors (the columns), leveraging the linearity property that preserves addition and scalar multiplication.

Conditions: The matrix is a 3x3 transformation matrix.; The input vector has coordinates $(x, y, z)$.; The transformation is linear.

Matrix-vector multiplication in 3D works by scaling the columns of the matrix by the corresponding coordinates of the input vector and summing the results. The coordinates $(x, y, z)$ act as scalar multipliers for the transformed basis vectors (the columns), leveraging the linearity property that preserves addition and scalar multiplication.

Conditions: The matrix is a 3x3 transformation matrix.; The input vector has coordinates $(x, y, z)$.; The transformation is linear.