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.

1. Determine where <1, 0> and <0, 1> are mapped. 2. Place the transformed <1, 0> as the first column and the transformed <0, 1> as the second column of the matrix.

1. Determine the angle of rotation. 2. Construct the rotation matrix. 3. Multiply the rotation matrix by the vector.

1. Find the determinant of the transformation matrix. 2. Take the absolute value of the determinant. 3. Multiply the original area by the absolute value of the determinant.

1. Find the matrices for each individual transformation. 2. Multiply the matrices in the correct order (right to left).

1. Calculate the determinant of the matrix. 2. If the determinant is non-zero, the matrix has an inverse.

1. Find the composite transformation matrix. 2. Multiply the composite matrix by the vector.

1. Find the matrix for reflection over the x-axis. 2. Find the matrix for a 45-degree rotation. 3. Multiply the rotation matrix by the reflection matrix.

1. Calculate the determinant (ad-bc). 2. Swap 'a' and 'd', negate 'b' and 'c'. 3. Divide each element by the determinant.

1. Create a diagonal matrix. 2. Put 2 in the top-left entry and 3 in the bottom-right entry. 3. The matrix is $\begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix}$.

1. Multiply the two matrices together. 2. If the result is the identity matrix, then they are inverses.

A function that moves and stretches vectors, keeping lines straight and the origin fixed.

A matrix that represents a linear transformation, dictating how vectors are transformed.

The transformed vector resulting from multiplying a vector by a transformation matrix.

A matrix that rotates a vector by a specific angle, typically counterclockwise.

For a 2x2 matrix, it's the value calculated as $a_{11}a_{22} - a_{12}a_{21}$.

The absolute value of the determinant, indicating the scaling of areas under a transformation.

Applying one linear transformation after another, where the output of the first becomes the input of the second.

A transformation that 'undoes' the effect of another transformation.

A matrix that, when multiplied by the original matrix, results in the identity matrix.

A matrix that, when multiplied by any matrix, leaves the matrix unchanged. Denoted as I.