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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.

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.

Geometrically, the determinant of $a 2\times 2$ matrix represents the factor by which the linear transformation scales areas. Specifically, it is the signed area of the parallelogram formed by the matrix's column vectors.

Conditions: The matrix is $2\times 2$.; The transformation is linear.; Use ordinary Euclidean area in standard orthonormal coordinates.

If the columns of a transformation matrix are linearly dependent, it means the transformation collapses the space into a lower dimension. For example, a 2D plane might be squashed into a 1D line or a single point.

Conditions: The matrix represents a linear transformation.; Columns are linearly dependent.

The matrix representation is $\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$. In this transformation, the standard basis vector $\hat{i}$ remains fixed at $[1, 0]^T$, while $\hat{j}$ slides diagonally to $[1, 1]^T$.

Conditions: Horizontal shear keeping $\hat{i}$ fixed.; $\hat{j}$ slides diagonally to $[1, 1]^T$.

Valid linear transformations keep grid lines straight, parallel, and evenly spaced. Nonlinear distortions involve bending or curving of these lines.

Conditions: Observing the movement of a 2D coordinate grid.; Checking for straightness, parallelism, and equal spacing of lines.

The matrix is $\begin{bmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ -1 & 0 & 0 \end{bmatrix}$. This is derived by tracking the standard basis vectors: $\hat{i}$ moves to $(0, 0, -1)$, $\hat{j}$ remains stationary at $(0, 1, 0)$, and $\hat{k}$ swings to $(1, 0, 0)$.

Conditions: Rotation is 90 degrees around the y-axis.; Working in a right-handed 3D coordinate system.