AP®︎ Precalculus: Topic 4.13 Practice Test

Essential Knowledge: 4.13.A.1

The linear transformation mapping $\langle x, y\rangle$ to $\langle a_{11}x + a_{12}y, a_{21}x + a_{22}y\rangle$ is associated with the matrix $\begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$.

Essential Knowledge: 4.13.A.2

The mapping of the unit vectors in a linear transformation provides valuable information for determining the associated matrix.

Essential Knowledge: 4.13.A.3

The matrix $\begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}$ is associated with a linear transformation of vectors that rotates every vector an angle $\theta$ counterclockwise about the origin.

Essential Knowledge: 4.13.A.4

The absolute value of the determinant of a $2 \times 2$ transformation matrix gives the magnitude of the dilation of regions in $\mathbb{R}^2$ under the transformation.

Essential Knowledge: 4.13.B.1

The composition of two linear transformations is a linear transformation.

Essential Knowledge: 4.13.B.2

The matrix associated with the composition of two linear transformations is the product of the matrices associated with each linear transformation.

Essential Knowledge: 4.13.C.1

Two linear transformations are inverses if their composition maps any vector to itself.

Essential Knowledge: 4.13.C.2

If a linear transformation, $L$, is given by $L(\vec{v}) = A\vec{v}$, then its inverse transformation is given by $L^{-1}(\vec{v}) = A^{-1}\vec{v}$, where $A^{-1}$ is the inverse of the matrix $A$.