What is the missing middle column if the shown method is carried to completion?

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.

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.