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In this context, $A \circ B$ denotes the transformation obtained by applying B first and then applying A to the result. The notation follows the convention where the rightmost transformation acts first on the input vector.

Conditions: A and B are treated as transformations on the same space, here three-dimensional space.; The output of B must be an input acceptable to A.

The missing middle column is $\begin{bmatrix} 0 \\ -16 \\ 9 \end{bmatrix}$. This is obtained by applying the transformation A to the middle column of B, which is $\begin{bmatrix} 0 \\ 2 \\ 3 \end{bmatrix}$.

Conditions: A and B are the specific 3x3 matrices shown in the video.; The method of column-wise composition is used.

During a 90-degree rotation around the y-axis, $\hat{i}$ moves to the negative z-axis at $(0, 0, -1)$, $\hat{j}$ remains stationary on the y-axis at $(0, 1, 0)$, and $\hat{k}$ swings to the positive x-axis at $(1, 0, 0)$.

Conditions: Rotation angle is exactly 90 degrees; Axis of rotation is the y-axis; Right-handed coordinate system convention assumed for sign determination

$A 2\times 2$ transformation matrix maps the standard basis vectors to its own columns. Specifically, the first column of the matrix is the image of the vector $[1, 0]^T$, and the second column is the image of the vector $[0, 1]^T$.

Conditions: The matrix is $2\times 2$.; Working in standard Cartesian coordinates.

The determinant scales the area of any measurable planar figure, not just squares. The absolute value of the determinant acts as a uniform area scaling factor for all regions under the linear transformation.

Conditions: The transformation is linear.; The figure is measurable and has finite area.; Use ordinary Euclidean area in standard orthonormal coordinates.

The matrix representation is $\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$. This is derived by observing where the standard basis vectors land: $\hat{i}$ rotates to $[0, 1]^T$ and $\hat{j}$ rotates to $[-1, 0]^T$.

Conditions: Rotation is 90 degrees counterclockwise.; Working in a 2D plane with standard basis vectors.

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 $\hat{j} = [0, 1]^T$.

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.

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-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.

Multiplying two 3x3 matrices represents applying their respective spatial transformations sequentially. The rightmost matrix acts first on the initial space, and the leftmost matrix acts on the already transformed result.

Conditions: Both matrices are 3x3 transformation matrices.; The operation is standard matrix multiplication.