AP®︎ Precalculus: Topic 4.12 Practice Test

Essential Knowledge: 4.12.A.1

A linear transformation is a function that maps an input vector to an output vector such that each component of the output vector is the sum of constant multiples of the input vector components.

Essential Knowledge: 4.12.A.2

A linear transformation will map the zero vector to the zero vector.

Essential Knowledge: 4.12.A.3

A single vector in $\mathbb{R}^{2}$ can be expressed as a $2 \times 1$ matrix. A set of $n$ vectors in $\mathbb{R}^{2}$ can be expressed as a $2 \times n$ matrix.

Essential Knowledge: 4.12.A.4

For a linear transformation, $L$, from $\mathbb{R}^{2}$ to $\mathbb{R}^{2}$, there is a unique $2 \times 2$ matrix, $A$, such that $L(\vec{v}) = A\vec{v}$ for vectors in $\mathbb{R}^{2}$. Conversely, for a given $2 \times 2$ matrix, $A$, the function $L(\vec{v}) = A\vec{v}$ is a linear transformation from $\mathbb{R}^{2}$ to $\mathbb{R}^{2}$.

Essential Knowledge: 4.12.A.5

Multiplication of a $2 \times 2$ transformation matrix, $A$, and a $2 \times n$ matrix of $n$ input vectors gives a $2 \times n$ matrix of the $n$ output vectors for the linear transformation $L(\vec{v}) = A\vec{v}$.