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To construct the 3x3 matrix, observe where the three standard basis vectors ($\hat{i}$, $\hat{j}$, and $\hat{k}$) land after the transformation. Record the final coordinates of these transformed vectors as column vectors.

Conditions: Working in three-dimensional Cartesian space.; The transformation is linear.

The origin stays fixed because linearity requires $T(av+bw)=aT(v)+bT(w)$. If we set $v=0$ and $w=0$ (or simply consider the zero vector), $T(0) = T(0\cdot v) = 0\cdot T(v) = 0$.

Conditions: The transformation is linear.; Applying the definition of linearity to the zero vector.

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.

Matrix-vector multiplication is reframed as scaling the transformed basis vectors (which are the columns of the matrix) by the input vector's components and summing them. This constructs the final position within the skewed coordinate system defined by the matrix, rather than just following a tedious arithmetic recipe.

Conditions: Input and output coordinates use the fixed standard basis.; The matrix represents an active linear map moving vectors.

We create a $2 \times 2$ matrix where the first column records the coordinates of where $\hat{i}$ lands, and the second column records the coordinates of where $\hat{j}$ lands. This matrix fully represents the linear transformation.

Conditions: Working in 2D space.; Knowing the images of $\hat{i}$ and $\hat{j}$.

The j-th column of the composed matrix $A \circ B$ is the image under A of the j-th column of B. To find a missing column, identify the corresponding column in B, treat it as an input vector, and apply the transformation A to it.

Conditions: A is a linear transformation represented by a matrix.; Columns are read as vectors in the domain of A.

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 $[1, 1]^T$.

To find the area of the transformed region, multiply the original area by the absolute value of the determinant of the transformation matrix. The formula is: $\text{New Area} = \text{Old Area} \cdot |\det(A)|$.

Conditions: The transformation is linear and represented by $a 2\times 2$ matrix.; The original area is known.; Use ordinary Euclidean area in standard orthonormal coordinates.

When multiplying two matrices $A$ and $B$ to form $AB$, the transformation represented by the rightmost matrix $B$ acts first on the initial space, followed by the leftmost matrix $A$ acting on the already transformed result.

Conditions: Performing matrix multiplication $AB$; Matrices represent linear spatial transformations; Vectors are treated as column vectors

A 3D linear transformation is fully determined by tracking where the standard basis vectors ($\hat{i}$, $\hat{j}$, and $\hat{k}$) land. The coordinates of these three transformed vectors are recorded as column vectors to form a 3x3 matrix.

Conditions: Working in three-dimensional Cartesian space; Using the standard basis vectors aligned with x, y, and z axes; The transformation is linear (preserves grid lines parallel/evenly spaced and fixes origin)

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.