Linear Transformations and Matrices - AP PreCalculus Study Guide

The Core Idea: Linear Transformations and Matrices

A linear transformation is a fundamental concept that describes a specific type of function for manipulating geometric objects in a plane. Imagine taking a point or a vector and moving it to a new location according to a consistent rule; this process is a transformation. The core idea of this topic is that these geometric transformations—which can involve actions like stretching, shearing, or rotating—can be represented and calculated using algebra through the use of a 2x2 matrix.

This connection is powerful because it translates a visual, geometric action into a precise, computational procedure. A 2x2 matrix acts as the "engine" of the transformation. When this matrix is multiplied by a vector representing a point's coordinates, the result is a new vector that gives the coordinates of the transformed point. This allows us to systematically determine how any point, or an entire collection of points forming a shape, will change under a given linear transformation.

Key Formulas and Representations

The central process in this topic is applying a transformation matrix to a point. A point $(x,y)$ in the plane is represented as a 2x1 column vector, and the linear transformation is represented by a 2x2 matrix.

The Transformation Equation

The transformation of a point $(x,y)$ to a new point $(x^{\prime },y^{\prime })$ by a matrix $A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ is defined by the following matrix equation:

$$\left(\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right)=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)$$

• $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ is the transformation matrix.

• $\left(\begin{array}{c}x\\ y\end{array}\right)$ is the original vector (representing the initial point).

• $\left(\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right)$ is the image vector (representing the transformed point).

The Coordinate Formulas

By performing the matrix multiplication, we can express the coordinates of the new point $(x^{\prime },y^{\prime })$ directly in terms of the original coordinates $(x,y)$ and the elements of the transformation matrix:

$$x^{\prime }=ax+by$$

$$y^{\prime }=cx+dy$$

This shows that each new coordinate is a linear combination of the original coordinates.

Understanding the Matrix-Vector Product

The concept of a linear transformation hinges on the operation of matrix-vector multiplication. It's crucial to understand that this is not just an arbitrary calculation; it is the mechanism that maps every input vector to its corresponding output vector.

A linear transformation, $T$, is a function that takes a vector $\vec{v}$ as input and produces a new vector $T(\vec{v})$ as output. The 2x2 matrix is the algebraic representation of the function $T$.

When we calculate $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)$, we are essentially applying the rule of the transformation.

• The top element of the resulting vector, $x^{\prime }$, is found by taking the dot product of the first row of the matrix with the column vector: $(a,b)\cdot (x,y)=ax+by$.

• The bottom element of the resulting vector, $y^{\prime }$, is found by taking the dot product of the second row of the matrix with the column vector: $(c,d)\cdot (x,y)=cx+dy$.

This process ensures that every point in the plane is transformed in a consistent and linear fashion, as dictated by the four values within the transformation matrix.

Core Concepts & Rules

• Transformation as a Function: A linear transformation is a function, $T$, that maps an input vector, $\vec{v}$, to an output vector, $T(\vec{v})$.

• Matrix Representation: Any linear transformation of a two-dimensional plane can be fully described by a single 2x2 matrix.

• Applying the Transformation: To find the result of a transformation on a specific point, you must represent the point as a 2x1 column vector and multiply it by the 2x2 transformation matrix.

• Order of Multiplication: The transformation matrix must be on the left, and the column vector representing the point must be on the right. The product is $A\vec{v}$, not $\vec{v}A$.

• Output Interpretation: The result of this multiplication is a new 2x1 column vector, where the top element is the new x-coordinate ($x^{\prime }$) and the bottom element is the new y-coordinate ($y^{\prime }$).

• Coordinate Calculation Rule: For a transformation matrix $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ and a point $(x,y)$, the transformed point $(x^{\prime },y^{\prime })$ is always calculated as $(ax+by,cx+dy)$.

Step-by-Step Example 1: Transforming a Single Point

Problem: A linear transformation $T$ is represented by the matrix $A=\left(\begin{array}{cc}3& -2\\ 5& 1\end{array}\right)$. Find the image of the point $P(-4,6)$ under this transformation.

Step 1: Represent the Point as a Column Vector

The point $P(-4,6)$ is represented by the column vector $\vec{v}=\left(\begin{array}{c}-4\\ 6\end{array}\right)$.

Step 2: Set Up the Matrix Multiplication

The image of the point, which we will call $P^{\prime }$, is found by calculating the product of the matrix $A$ and the vector $\vec{v}$.

$$\left(\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right)=A\vec{v}=\left(\begin{array}{cc}3& -2\\ 5& 1\end{array}\right)\left(\begin{array}{c}-4\\ 6\end{array}\right)$$

Step 3: Perform the Multiplication

Calculate the new x-coordinate ($x^{\prime }$) using the first row of the matrix and the new y-coordinate ($y^{\prime }$) using the second row.

• For the new x-coordinate, $x^{\prime }$:

  $$x^{\prime }=(3)(-4)+(-2)(6)=-12-12=-24$$

• For the new y-coordinate, $y^{\prime }$:

  $$y^{\prime }=(5)(-4)+(1)(6)=-20+6=-14$$

Step 4: State the Final Coordinates

The resulting vector is $\left(\begin{array}{c}-24\\ -14\end{array}\right)$. Therefore, the image of the point $P(-4,6)$ is $P^{\prime }(-24,-14)$.

Step-by-Step Example 2: Transforming the Vertices of a Shape

Problem: A triangle in the coordinate plane is defined by the vertices $A(1,2)$, $B(4,5)$, and $C(0,-3)$. Find the coordinates of the vertices of the transformed triangle, $△A^{\prime }{B}^{\prime }{C}^{\prime }$, after applying the linear transformation represented by the matrix $T=\left(\begin{array}{cc}0& 1\\ -2& 1\end{array}\right)$.

Step 1: Transform Each Vertex Individually

To find the transformed triangle, we must apply the transformation matrix to the column vector representation of each vertex.

Step 2: Calculate the Coordinates of A'

• Set up the multiplication for vertex $A(1,2)$:

  $$\left(\begin{array}{c}{x}_A^{\prime }\\ y_A^{\prime }\end{array}\right)=\left(\begin{array}{cc}0& 1\\ -2& 1\end{array}\right)\left(\begin{array}{c}1\\ 2\end{array}\right)$$

• Perform the calculation:

  $$x_A^{\prime }=(0)(1)+(1)(2)=2$$

  $$y_A^{\prime }=(-2)(1)+(1)(2)=-2+2=0$$

• The coordinates of the transformed vertex are $A^{\prime }(2,0)$.

Step 3: Calculate the Coordinates of B'

• Set up the multiplication for vertex $B(4,5)$:

  $$\left(\begin{array}{c}{x}_B^{\prime }\\ y_B^{\prime }\end{array}\right)=\left(\begin{array}{cc}0& 1\\ -2& 1\end{array}\right)\left(\begin{array}{c}4\\ 5\end{array}\right)$$

• Perform the calculation:

  $$x_B^{\prime }=(0)(4)+(1)(5)=5$$

  $$y_B^{\prime }=(-2)(4)+(1)(5)=-8+5=-3$$

• The coordinates of the transformed vertex are $B^{\prime }(5,-3)$.

Step 4: Calculate the Coordinates of C'

• Set up the multiplication for vertex $C(0,-3)$:

  $$\left(\begin{array}{c}{x}_C^{\prime }\\ y_C^{\prime }\end{array}\right)=\left(\begin{array}{cc}0& 1\\ -2& 1\end{array}\right)\left(\begin{array}{c}0\\ -3\end{array}\right)$$

• Perform the calculation:

  $$x_C^{\prime }=(0)(0)+(1)(-3)=-3$$

  $$y_C^{\prime }=(-2)(0)+(1)(-3)=0-3=-3$$

• The coordinates of the transformed vertex are $C^{\prime }(-3,-3)$.

Step 5: State the Final Answer

The vertices of the transformed triangle are $A^{\prime }(2,0)$, $B^{\prime }(5,-3)$, and $C^{\prime }(-3,-3)$.

Using Your Calculator

A graphing calculator is an efficient tool for performing the matrix multiplication required to find a transformed point, especially when the numbers are not integers.

Goal: Use a calculator to solve Example 1: Find the image of $P(-4,6)$ under the transformation given by $A=\left(\begin{array}{cc}3& -2\\ 5& 1\end{array}\right)$.

TI-84 Style Instructions:

1. Enter the Transformation Matrix:

   • Press [2nd]$then$[x⁻¹]to open the MATRIX menu. - Navigate to the `EDIT` tab. - Select `1: [A]` and press `[ENTER]`. - Set the dimensions to $2x2 by pressing $2$[ENTER]$2$[ENTER]`.

   • Enter the elements of the matrix: $3$[ENTER]$-2$[ENTER]$5$[ENTER]$1$[ENTER].

2. Enter the Point's Column Vector:

   • Press [2nd]$then$[x⁻¹]to open the MATRIX menu again. - Navigate to `EDIT` and select `2: [B]`. - Set the dimensions to $2x1 by pressing $2$[ENTER]$1$[ENTER]`.

   • Enter the coordinates of the point: $-4$[ENTER]$6$[ENTER].

3. Perform the Multiplication:

   • Press [2nd]$then$[MODE]$to‘QUIT‘tothehomescreen.-Press‘[2nd]$ then $[x^{-1}]$ to open the MATRIX menu. Under the NAMEStab, select1: [A]and press[ENTER]`.

   • Press the multiplication key $[\ast ]$.

   • Open the MATRIX menu again, select 2: [B] under NAMES, and press [ENTER].

   • Your screen should now display $[A]\ast [B]$.

4. Get the Result:

   • Press [ENTER]. The calculator will display the resulting 2x1 matrix:

     $[[-24][-14]]$

   • This corresponds to the transformed point $P^{\prime }(-24,-14)$.

AP Exam Quick Hit

Common Question Types

• Direct Application: You will be given a 2x2 matrix and the coordinates of a single point. The question will ask for the coordinates of the transformed point.

  • Example: "What are the coordinates of the point $(2,-5)$ after it is transformed by the matrix $\left(\begin{array}{cc}1& 0\\ 4& -1\end{array}\right)$?"

• Transforming a Shape: You will be given the vertices of a polygon (e.g., a triangle or quadrilateral) and a transformation matrix. The question will ask for the coordinates of one or all of the new vertices.

  • Example: "A square has vertices at (0,0), (1,0), (1,1), and (0,1). If the square is transformed by the matrix $\left(\begin{array}{cc}2& 1\\ 1& 2\end{array}\right)$, what is the new coordinate of the vertex that was originally at (1,1)?"

• Finding Coordinates in Terms of Variables: You will be given a generic point $(x,y)$ and a matrix, and asked to find a formula for the transformed coordinates.

  • Example: "A point $(x,y)$ is transformed by the matrix $\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$. Find the coordinates of the transformed point $(x^{\prime },y^{\prime })$ in terms of $x$ and $y$."

Common Mistakes

• Incorrect Order of Multiplication: Multiplying the vector by the matrix ($\vec{v}A$) instead of the matrix by the vector ($A\vec{v}$). Matrix multiplication is not commutative, and this will lead to an incorrect answer or a dimension mismatch error. Always place the 2x2 matrix on the left.

• Using a Row Vector: Representing the point $(x,y)$ as a 1x2 row vector $\left(\begin{array}{cc}x& y\end{array}\right)$ instead of a 2x1 column vector $\left(\begin{array}{c}x\\ y\end{array}\right)$. You cannot multiply a 2x2 matrix by a 1x2 vector.

• Arithmetic Errors: Making simple sign errors or calculation mistakes during the multiplication and addition steps. For example, in the calculation $(5)(-4)+(1)(6)$, a student might incorrectly calculate $-20+6$ as $-26$.

• Mixing Up Rows: When calculating the new y-coordinate ($y^{\prime }=cx+dy$), accidentally using the numbers from the first row of the matrix ($a$ and $b$) instead of the second row ($c$ and $d$).