**Identity transformation:** This transformation leaves the vector unchanged. The identity matrix has ones on the diagonal and zeros elsewhere. $ \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} * \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} x \\ y \end{bmatrix} $ **Inversion transformation:** This flips the signs of both $x,y$. Note that this is different from the matrix inverse. $ \begin{bmatrix} -1 & \phantom{..}0 \\ \phantom{..}0 & -1 \end{bmatrix} * \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} -x \\ -y \end{bmatrix} $ ![[matrix-inversion.png|center|250]] **Stretch transformation:** A matrix with an off-diagonal element in the upper corner amplifies the stretch along the $x$-axis. $ \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} * \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} x+y \\ y \end{bmatrix} $ ![[matrix-strech.png|center|250]] **Mirror transformation:** The original basis vectors switched their positions, which leads to the mirroring along a $45^\circ$ angle through the origin. $ \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} * \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} y \\ x \end{bmatrix} $ ![[matrix-mirror.png|center|300]]