Welcome to Linear Transformations!

In previous chapters, you learned what matrices are and how to perform basic operations with them. Now, we get to the "cool" part: using matrices to move things! If you have ever played a video game or used a photo editing app to rotate or resize an image, you have used linear transformations. In this chapter, we will explore how a simple grid of numbers can stretch, flip, or spin coordinates in a 2D plane.

Note: This topic is part of Unit 4, which means it is not assessed on the AP Exam. However, it is essential for anyone interested in physics, engineering, or computer science!

What is a Linear Transformation?

A linear transformation is a specific type of function that takes a vector (an input point) and "transforms" it into a new vector (an output point). For this course, we focus on 2D transformations, which move points from the \(xy\)-plane to new positions in the same plane.

For a transformation to be "linear," two things must stay true:

  • The origin \((0,0)\) must stay at \((0,0)\).
  • Grid lines must remain parallel and evenly spaced (no curving!).

The Matrix as a Machine

We represent a linear transformation using a \(2 \times 2\) matrix. When we multiply a matrix by a coordinate vector, the matrix acts like a "machine" that processes the input and produces a new output.

If we have a transformation matrix \(A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\) and an input vector \(\vec{v} = \begin{pmatrix} x \\ y \end{pmatrix}\), the transformed vector \(\vec{v'}\) is found by:

\(A\vec{v} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} ax + by \\ cx + dy \end{pmatrix}\)

Quick Review: Remember that you can review basic matrix multiplication in Topic 4.10 if the steps above feel a bit rusty!

The Secret Trick: Tracking \(\hat{i}\) and \(\hat{j}\)

Don't worry if the algebra looks messy! There is a much easier way to visualize what a matrix does. Every \(2 \times 2\) matrix is simply telling us where the two basis vectors land:

  • The first column tells us where the point \((1, 0)\) (often called \(\hat{i}\)) lands.
  • The second column tells us where the point \((0, 1)\) (often called \(\hat{j}\)) lands.

Example: If your matrix is \(\begin{pmatrix} 3 & 0 \\ 0 & 2 \end{pmatrix}\), it means \((1,0)\) moved to \((3,0)\) and \((0,1)\) moved to \((0,2)\). Everything else in the plane just follows their lead!

Key Takeaway:

The columns of a transformation matrix represent the images of the basis vectors \((1, 0)\) and \((0, 1)\).

Common 2D Transformations

Here are the most common "moves" you will see in AP Precalculus:

1. Dilations (Scaling)

A dilation stretches or shrinks the plane. If you want to stretch everything by a factor of \(k\), you use:

\(\begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}\)

Example: \(\begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}\) doubles the size of every shape.

2. Reflections

Reflections "flip" the plane over a specific line. You can find these by thinking about where \((1,0)\) and \((0,1)\) would go:

  • Reflect over the x-axis: \(\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\) (The \(x\) stays same, \(y\) flips sign).
  • Reflect over the y-axis: \(\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}\) (The \(y\) stays same, \(x\) flips sign).
  • Reflect over the line \(y = x\): \(\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\) (The \(x\) and \(y\) coordinates swap).

3. Rotations

To rotate the entire plane counterclockwise by an angle \(\theta\), we use the rotation matrix:

\(\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}\)

Did you know? This formula comes directly from the unit circle! If you rotate the point \((1,0)\) by \(\theta\), it lands at \((\cos\theta, \sin\theta)\), which is exactly the first column of our matrix.

Composition of Transformations

What if you want to reflect a shape and then rotate it? This is called a composition. In the world of matrices, composition is done through matrix multiplication.

Important Rule: Order matters! If you apply transformation \(A\) first and then transformation \(B\), you write the operation as \(B \times A\). You read it from right to left, just like function composition \(f(g(x))\).

Step-by-Step Composition:
1. Identify the matrix for the 1st transformation (let's call it \(M_1\)).
2. Identify the matrix for the 2nd transformation (let's call it \(M_2\)).
3. Multiply them: \(M_{total} = M_2 \times M_1\).
4. The result is a single matrix that does both jobs at once!

The Identity Transformation

There is one special matrix called the Identity Matrix, denoted by \(I\):

\(I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\)

This matrix is the "do-nothing" transformation. It keeps \((1,0)\) at \((1,0)\) and \((0,1)\) at \((0,1)\). Multiplying any vector by the identity matrix leaves the vector unchanged.

Common Mistakes to Avoid

  • Mixing up Columns: Remember, the first column is for the \(x\)-basis vector and the second is for the \(y\)-basis vector. Swapping them changes the transformation entirely!
  • Wrong Multiplication Order: Always place the matrix for the first transformation on the right when multiplying them together.
  • Degree vs. Radian: When using rotation matrices, check if your angle \(\theta\) is in degrees or radians. Most AP Precalculus problems default to radians.

Summary Table for Quick Reference

Dilation by factor \(k\): \(\begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}\)
Reflection (x-axis): \(\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\)
Reflection (y-axis): \(\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}\)
Rotation (CCW): \(\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}\)

Remember: You don't need to memorize every possible matrix. If you get stuck, just ask yourself: "Where would the point (1, 0) go? Where would the point (0, 1) go?" Write those two new points as columns, and you've found your matrix!