The transformed images of the standard basis vectors (unit vectors) become the columns of the transformation matrix.

The absolute value of the determinant indicates the scaling factor of the area under the transformation. The sign indicates orientation (reflection).

It involves applying one transformation after another. The matrix of the composition is the product of the individual transformation matrices.

It means there exists another transformation that 'undoes' the original transformation, mapping the output back to the original input.

Matrix multiplication is not commutative; therefore, the order in which transformations are applied matters. AB is not necessarily equal to BA.

A negative determinant indicates that the transformation includes a reflection (flip) in addition to any possible dilation.

The matrix is $\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}$ because it maps (1,0) to (-1,0) and (0,1) to (0,1).

The absolute value of the determinant gives the factor by which the area of any shape is scaled after the transformation.

A matrix has an inverse if and only if its determinant is non-zero. A zero determinant indicates that the transformation collapses space, making it impossible to 'undo'.

The inverse of a matrix, when multiplied by the original matrix, results in the identity matrix. This represents 'undoing' the transformation.

$\begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$

$\begin{bmatrix} cos(θ) & -sin(θ) \\ sin(θ) & cos(θ) \end{bmatrix}$

$x' = xcos(θ) - ysin(θ)$ , $y' = xsin(θ) + ycos(θ)$

$\begin{bmatrix} cos(-θ) & -sin(-θ) \\ sin(-θ) & cos(-θ) \end{bmatrix}$

For matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$, the determinant is $ad - bc$.

AB

L⁻¹(v) = A⁻¹v

$\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}$

$\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$

$\begin{bmatrix} k & 0 \\ 0 & k \end{bmatrix}$

Rotation: Preserves orientation, changes angle | Reflection: Reverses orientation (flips), maintains distance from axis.

Positive Determinant: Dilation (expansion), no reflection | Negative Determinant: Dilation and reflection.

Transformation: Maps input to output | Inverse: Maps output back to the original input, 'undoing' the transformation.

f(g(x)): Apply g first, then f | g(f(x)): Apply f first, then g. Order matters because matrix multiplication is not commutative.

Linear Transformation: Preserves lines, origin fixed | Non-Linear Transformation: May curve lines, origin may shift.

Scaling: Changes the size of an object | Shearing: Distorts the shape of an object, shifting points parallel to an axis.

Determinant of 1: Preserves area and orientation | Determinant of -1: Preserves area but reverses orientation (reflection).

Identity Matrix: Acts as '1' for matrix multiplication, leaves matrix unchanged | Zero Matrix: Acts as '0' for matrix addition, results in zero matrix upon multiplication.

Row Operations: Used to solve systems of equations, affect rows | Column Operations: Affect columns, related to linear combinations of column vectors.

Orthogonal Matrices: Columns (and rows) are orthonormal, preserve lengths and angles | Non-Orthogonal Matrices: Do not necessarily preserve lengths and angles.