4.13 Matrices as Functions

A 2 x 2 matrix can act like a function that takes an input vector and produces an output vector. The columns of that matrix tell you where the standard unit vectors land, so once you know how the transformation moves $\langle 1, 0\rangle$ and $\langle 0, 1\rangle$, you know the whole matrix.

Why This Matters for the AP Precalculus Exam

Unit 4 topics, including matrices as functions, are not assessed on the AP Precalculus exam. The exam covers Units 1, 2, and 3. Still, this topic is worth learning because it ties together vectors, matrix multiplication, determinants, and inverses into one clear idea: a matrix is a function on vectors.

The thinking you build here carries into later math and science. Connecting a rule like $\langle x, y\rangle \mapsto \langle a_{11}x + a_{12}y,\ a_{21}x + a_{22}y\rangle$ to a single matrix is exactly the kind of move-between-representations reasoning the course wants you to practice. If your school includes Unit 4, use technology to build matrices, multiply them, and find inverses, just as you would for other calculator-supported work.

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Key Takeaways

• The linear transformation that sends $\langle x, y\rangle$ to $\langle a_{11}x + a_{12}y,\ a_{21}x + a_{22}y\rangle$ matches the matrix $\begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$.
• The columns of the matrix are the images of the unit vectors $\langle 1, 0\rangle$ and $\langle 0, 1\rangle$, so mapping the unit vectors gives you the matrix.
• The rotation matrix $\begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}$ rotates every vector an angle $\theta$ counterclockwise about the origin.
• The absolute value of the determinant tells you how much the transformation scales the area of regions in $\mathbb{R}^2$.
• The matrix for a composition of two transformations is the product of the two matrices, and order matters.
• If $L(v) = Av$ and $A$ is invertible, the inverse transformation is $L^{-1}(v) = A^{-1}v$.

Connecting Linear Transformations and Matrices

The linear transformation mapping a vector $\langle x, y\rangle$ to a vector $\langle a_{11}x + a_{12}y,\ a_{21}x + a_{22}y\rangle$ is represented by the matrix $\begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$, a 2 x 2 matrix. This is called a transformation matrix, and it stores all the information about the transformation in the four coefficients $a_{11}$, $a_{12}$, $a_{21}$, and $a_{22}$.

When the vector $\langle x, y\rangle$ is multiplied by the matrix $\begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$, the result is the new vector $\langle a_{11}x + a_{12}y,\ a_{21}x + a_{22}y\rangle$. That new vector is the image of the original vector under the transformation.

Mapping the Unit Vectors

The mapping of the unit vectors gives you a fast way to find the matrix. In two dimensions, the unit vectors are $\langle 1, 0\rangle$ and $\langle 0, 1\rangle$, often called the standard basis vectors. The transformation sends each unit vector to a new vector, and those new vectors are the columns of the transformation matrix.

For example, if the transformation maps $\langle 1, 0\rangle$ to $\langle a_{11}, a_{21}\rangle$ and maps $\langle 0, 1\rangle$ to $\langle a_{12}, a_{22}\rangle$, then the transformation matrix is $\begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$. Track where the unit vectors land, write those results as columns, and you have the matrix.

Rotation Matrices

The matrix associated with rotating every vector an angle $\theta$ counterclockwise about the origin is the rotation matrix:

$\begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}$

When a vector $\langle x, y\rangle$ is multiplied by this matrix, the resulting vector $\langle x', y'\rangle$ is the image of the original vector under the rotation:

$x' = x\cos\theta - y\sin\theta$

$y' = x\sin\theta + y\cos\theta$

So $x'$ and $y'$ are the coordinates after rotating by an angle $\theta$ counterclockwise. The matrix stores everything about the rotation, including the angle $\theta$.

To rotate clockwise instead, replace $\theta$ with $-\theta$. Because $\cos(-\theta) = \cos\theta$ and $\sin(-\theta) = -\sin\theta$, the clockwise rotation matrix is $\begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix}$.

Determinants and Area Scaling

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

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

The absolute value of this number is the area-scaling factor. If $|\det A| = 2$, the transformation doubles the area of regions. If $|\det A| = \tfrac{1}{2}$, it cuts areas in half. The sign of the determinant tells you about orientation: a negative determinant means the transformation flips orientation, like a reflection, while still scaling area by the absolute value.

Compositions of Two Linear Transformations

The composition of two linear transformations is itself a linear transformation. A linear transformation takes a vector as input and produces a vector as output. When you compose two of them, the output of the first becomes the input of the second.

If $f$ maps a vector $x$ to a vector $y$, and $g$ maps $y$ to a vector $z$, then the composition is written $g(f(x))$ and it sends $x$ all the way to $z$.

The matrix associated with the composition of two linear transformations is the product of the matrices associated with each transformation. If $A$ is the matrix for $f$ and $B$ is the matrix for $g$, then the matrix for the composition that applies $f$ first and then $g$ is $BA$. Multiplying a vector by that product carries it through both transformations in order.

Order matters. In general $AB \neq BA$, so swapping the order of the matrices usually changes the result. Keep careful track of which transformation happens first.

Inverses of Linear Transformations

Two linear transformations are inverses if their composition maps any vector back to itself. An inverse transformation undoes the effect of the original. If $f$ maps $x$ to $y$ and $g$ maps $y$ back to $x$, then $g$ is the inverse of $f$, written $f^{-1}$.

If a linear transformation $L$ is given by $L(v) = Av$, then its inverse transformation is:

$L^{-1}(v) = A^{-1}v$

where $A^{-1}$ is the inverse of the matrix $A$. This follows directly from how matrix-vector multiplication works. The matrix $A$ encodes the transformation, and $A^{-1}$ reverses it:

$L^{-1}(Av) = A^{-1}(Av) = A^{-1}A\,v = Iv = v$

Here $I$ is the identity matrix, the matrix that leaves every vector unchanged. Applying $A^{-1}$ to the output $Av$ returns the original vector $v$.

Not every matrix has an inverse. A 2 x 2 matrix $A$ is invertible if and only if its determinant is not zero. When $\det A = 0$, the transformation collapses the plane onto a line or a point, so there is no way to undo it.

How to Use This on the AP Precalculus Exam

Problem Solving

• To find a transformation matrix, track where $\langle 1, 0\rangle$ and $\langle 0, 1\rangle$ go. Write each image as a column, and you have the matrix.
• To rotate a vector counterclockwise by $\theta$, multiply by $\begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}$. For clockwise, use $-\theta$.
• To find how a transformation scales area, compute $|\det A|$. A negative determinant also flips orientation.
• To compose transformations, multiply the matrices in the right order. The transformation applied first sits on the right of the product.
• To undo a transformation $L(v) = Av$, use $L^{-1}(v) = A^{-1}v$, and remember it only exists when $\det A \neq 0$.

Using Technology

If your school covers Unit 4, practice building matrices, multiplying them, and finding inverses on a graphing calculator. These are the same calculator skills used elsewhere in the course, so getting comfortable with them helps your overall fluency.

Common Misconceptions

• The columns of the matrix, not the rows, are the images of the unit vectors. $\langle 1, 0\rangle$ maps to the first column and $\langle 0, 1\rangle$ maps to the second column.
• Order matters when composing. The matrix for "do $f$ first, then $g$" is $BA$, not $AB$. Matrix multiplication is generally not commutative.
• The determinant scales area, not length. $|\det A|$ tells you the area-scaling factor for regions, not how much a single vector stretches.
• A negative determinant does not mean the area shrinks. The area scales by the absolute value. The negative sign signals a flip in orientation, like a reflection.
• Not every matrix is invertible. If $\det A = 0$, the transformation squashes the plane to a line or point and cannot be reversed, so $A^{-1}$ does not exist.
• The rotation matrix rotates about the origin. It rotates every vector around the origin by $\theta$, not around some other point.

Related AP Precalculus Guides

• 4.9 Vector-Valued Functions
• 4.11 The Inverse and Determinant of a Matrix
• 4.5 Implicitly Defined Functions
• 4.8 Vectors
• 4.12 Linear Transformations and Matrices
• 4.4 Parametrically Defined Circles and Lines