How is matrix subtraction defined for matrices of the same dimensions?

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.

$A 2\times 2$ 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.

Each column of $a 2\times 2$ matrix can be read as the coordinates of a vector in the 2D plane. For the matrix $A = \begin{bmatrix} 3 & 1 \\ 1 & 2 \end{bmatrix}$, the first column $\begin{bmatrix} 3 \\ 1 \end{bmatrix}$ represents the vector from the origin to the point (3,1), and the second column $\begin{bmatrix} 1 \\ 2 \end{bmatrix}$ represents the vector from the origin to the point (1,2).

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

Yes, matrix addition is commutative. For any two matrices A and B with the same dimensions, $A + B$ equals $B + A$.

Conditions: Matrices A and B must have identical dimensions.; Entries are real numbers.

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.