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AP Course

AP Precalculus

Study nine focused AP Precalculus units through topic lessons, original graph examples, quizzes, and practice sets.

Follow the College Board topic sequence across polynomial, rational, exponential, logarithmic, trigonometric, polar, parametric, vector, and matrix functions.

Lessons
Change in TandemRates of ChangeRates of Change in Linear and Quadratic FunctionsPolynomial Functions and Rates of ChangePolynomial Functions and Complex ZerosPolynomial Functions and End BehaviorRational Functions and End BehaviorRational Functions and ZerosRational Functions and Vertical AsymptotesRational Functions and HolesEquivalent Representations of Polynomial and Rational ExpressionsTransformations of FunctionsFunction Model Selection and Assumption ArticulationFunction Model Construction and ApplicationChange in Arithmetic and Geometric SequencesChange in Linear and Exponential FunctionsExponential FunctionsExponential Function ManipulationExponential Function Context and Data ModelingCompeting Function Model ValidationComposition of FunctionsInverse FunctionsLogarithmic ExpressionsInverses of Exponential FunctionsLogarithmic FunctionsLogarithmic Function ManipulationExponential and Logarithmic Equations and InequalitiesLogarithmic Function Context and Data ModelingSemi-log PlotsPeriodic PhenomenaSine, Cosine, and TangentSine and Cosine Function ValuesSine and Cosine Function GraphsSinusoidal FunctionsSinusoidal Function TransformationsSinusoidal Function Context and Data ModelingThe Tangent FunctionInverse Trigonometric FunctionsTrigonometric Equations and InequalitiesThe Secant, Cosecant, and Cotangent FunctionsEquivalent Representations of Trigonometric FunctionsTrigonometry and Polar CoordinatesPolar Function GraphsRates of Change in Polar FunctionsParametric FunctionsParametric Functions Modeling Planar MotionParametric Functions and Rates of ChangeParametrically Defined Circles and LinesImplicitly Defined FunctionsConic SectionsParametrization of Implicitly Defined FunctionsVectorsVector-Valued FunctionsMatricesThe Inverse and Determinant of a MatrixLinear Transformations and MatricesMatrices as FunctionsMatrices Modeling Contexts
Quizzes
Practice Problems Open a unit to choose from 58 topic-aligned practice sets.

Unit 9 · Topic 4.13 · Optional

Matrices as Functions

Move fluently among a linear transformation rule, its matrix, its action on basis vectors, compositions of transformations, and inverse transformations.

Learning Goals

  • Construct a transformation matrix from coordinate rules or basis-vector images.
  • Use the general rotation matrix to rotate vectors through any angle.
  • Interpret the determinant as an area-dilation factor.
  • Represent a composition with matrix multiplication in the correct order.
  • Explain why transformation composition is generally not commutative.
  • Construct and verify an inverse linear transformation.

1. A Matrix Represents a Function

Every \(2\times2\) matrix A defines a linear function from \(\mathbb R^2\) to \(\mathbb R^2\):

\[L_A(\mathbf v)=A\mathbf v.\]

The same transformation can be represented verbally, with coordinate rules, as a matrix, or by showing how it moves geometric objects. These representations must produce the same output for every input vector.

2. Translate a Coordinate Rule into a Matrix

The rule

\[L(x,y)=(ax+by,\;cx+dy)\]

corresponds to

\[A=\begin{bmatrix}a&b\\c&d\end{bmatrix},\qquad L\begin{pmatrix}x\\y\end{pmatrix} =A\begin{bmatrix}x\\y\end{bmatrix}.\]

Coefficients in the first output form row 1; coefficients in the second output form row 2.

3. The Columns Are Basis-Vector Images

Because \(\mathbf i=\langle1,0\rangle\) and \(\mathbf j=\langle0,1\rangle\),

\[A=\begin{bmatrix}\vert&\vert\\L(\mathbf i)&L(\mathbf j)\\\vert&\vert\end{bmatrix}.\]

The first column is \(L(\mathbf i)\), and the second is \(L(\mathbf j)\). Since every vector is \(x\mathbf i+y\mathbf j\), these two images determine the entire linear transformation.

4. Basis Vectors Determine the Plane

Images of the standard basis vectorsThe standard unit vectors are shown with two transformed image vectors that become the columns of a matrix.ijL(i)L(j)
The image of i becomes column 1, and the image of j becomes column 2.

5. Build a Matrix from Two Images

Suppose

\[L(\mathbf i)=\begin{bmatrix}2\\1\end{bmatrix}, \qquad L(\mathbf j)=\begin{bmatrix}-1\\3\end{bmatrix}.\]

Place these vectors as columns:

\[A=\begin{bmatrix}2&-1\\1&3\end{bmatrix}.\]

Then \(L(4,2)=4L(\mathbf i)+2L(\mathbf j)=\langle6,10\rangle\), matching \(A\begin{bmatrix}4\\2\end{bmatrix}\).

6. Rotation through Any Angle

A counterclockwise rotation through angle \(\theta\) about the origin is represented by

\[R_\theta= \begin{bmatrix} \cos\theta&-\sin\theta\\ \sin\theta&\cos\theta \end{bmatrix}.\]

Its columns are the rotated images of \(\mathbf i\) and \(\mathbf j\). Because \(\det(R_\theta)=\cos^2\theta+\sin^2\theta=1\), rotation preserves area and orientation.

7. Rotation Example

At \(\theta=\pi/2\),

\[R_{\pi/2}=\begin{bmatrix}0&-1\\1&0\end{bmatrix}, \qquad R_{\pi/2}\begin{bmatrix}3\\-2\end{bmatrix} =\begin{bmatrix}2\\3\end{bmatrix}.\]

The vector turns \(90^\circ\) counterclockwise and keeps magnitude \(\sqrt{13}\).

8. Determinant as a Region-Dilation Factor

ValueArea effectOrientation
\(\det A>0\)Scale by \(|\det A|\)Preserved
\(\det A<0\)Scale by \(|\det A|\)Reversed
\(\det A=0\)Collapse to zero areaTransformation is not invertible

If a region initially has area S, its image has area \(|\det A|S\).

9. Composition Becomes Matrix Multiplication

Let \(L_A(\mathbf v)=A\mathbf v\) and \(L_B(\mathbf v)=B\mathbf v\). Applying A first and B second gives

\[(L_B\circ L_A)(\mathbf v) =L_B(A\mathbf v) =B(A\mathbf v) =(BA)\mathbf v.\]

The matrix nearest the input acts first. Function-composition order and matrix-product order both read from right to left.

10. Worked Composition Example

First reflect across the x-axis with \(F=\begin{bmatrix}1&0\\0&-1\end{bmatrix}\), then apply the horizontal shear \(S=\begin{bmatrix}1&2\\0&1\end{bmatrix}\). The composite matrix is

\[SF= \begin{bmatrix}1&2\\0&1\end{bmatrix} \begin{bmatrix}1&0\\0&-1\end{bmatrix} = \begin{bmatrix}1&-2\\0&-1\end{bmatrix}.\]

For \(\mathbf v=\langle3,1\rangle\), reflection gives \(\langle3,-1\rangle\), then shearing gives \(\langle1,-1\rangle\). The product matrix gives the same result directly.

11. Reversing the Order Changes the Function

Shearing first and reflecting second produces

\[FS= \begin{bmatrix}1&2\\0&-1\end{bmatrix}.\]

Now \(\langle3,1\rangle\mapsto\langle5,-1\rangle\), not \(\langle1,-1\rangle\). Thus \(L_S\circ L_F\ne L_F\circ L_S\), just as \(SF\ne FS\).

12. Composition and Area Factors

\[\det(BA)=\det(B)\det(A).\]

Therefore, area dilation factors multiply under composition. A scale by 2 followed by a scale by 3 changes area by a factor of 6. If either determinant is 0, the composite also collapses area and cannot be inverted.

13. Inverse Transformations

Two transformations are inverses when both compositions return every vector unchanged:

\[L^{-1}(L(\mathbf v))=\mathbf v, \qquad L(L^{-1}(\mathbf v))=\mathbf v.\]

If \(L(\mathbf v)=A\mathbf v\) and \(\det A\ne0\), then

\[L^{-1}(\mathbf v)=A^{-1}\mathbf v.\]

A transformation with determinant 0 has merged distinct directions or points, so the lost information cannot be uniquely recovered.

14. Verify an Inverse Function

For

\[A=\begin{bmatrix}2&1\\1&1\end{bmatrix}, \qquad A^{-1}=\begin{bmatrix}1&-1\\-1&2\end{bmatrix},\]

the input \(\mathbf v=\langle4,3\rangle\) maps to

\[A\begin{bmatrix}4\\3\end{bmatrix} =\begin{bmatrix}11\\7\end{bmatrix},\qquad A^{-1}\begin{bmatrix}11\\7\end{bmatrix} =\begin{bmatrix}4\\3\end{bmatrix}.\]

Matrix multiplication also verifies \(AA^{-1}=A^{-1}A=I\).

15. AP Workflow and Common Errors

  1. Translate coordinate coefficients into matrix rows.
  2. Alternatively, place \(L(\mathbf i)\) and \(L(\mathbf j)\) as matrix columns.
  3. For rotation, substitute the angle into the standard matrix.
  4. For a composition, write the matrix acting first nearest the vector.
  5. Multiply matrices before applying the composite to inputs.
  6. Use \(|\det A|\) for area dilation and the determinant sign for orientation.
  7. Use \(A^{-1}\) only when \(\det A\ne0\), then verify both compositions.
  • Do not place basis-vector images as rows.
  • Do not reverse the matrix order in a composition.
  • Do not interchange the two sine signs in the rotation matrix.
  • Do not use a negative determinant as a negative area.
  • Do not assume transformations commute.
  • Do not claim an inverse exists when the determinant is 0.

Key Takeaways

  • View matrix multiplication as a function and connect matrix products with composition order.
  • Core relationship: \(T_A(\mathbf x)=A\mathbf x,\quad(T_B\circ T_A)(\mathbf x)=BA\mathbf x\)
  • Error check: Do not reverse the matrix product when translating a composition.
Checkpoint · Topic 4.13
  1. Write the matrix associated with \(L(x,y)=(3x-2y,5x+y)\).
  2. Construct A when \(L(\mathbf i)=\langle1,4\rangle\) and \(L(\mathbf j)=\langle-2,3\rangle\), then find \(L(2,-1)\).
  3. Use \(R_{\pi/3}\) to rotate \(\langle2,0\rangle\) counterclockwise.
  4. A reflection across the y-axis is followed by a \(90^\circ\) counterclockwise rotation. Find the composite matrix and apply it to \(\langle3,-1\rangle\).
  5. Find the inverse transformation associated with \(\begin{bmatrix}3&1\\2&1\end{bmatrix}\) and verify it on one nonzero vector.