#AP Pre-Calculus: Matrices as Functions - Your Night-Before Guide 🚀

Hey there! Let's make sure you're feeling super confident about matrices and linear transformations for your AP Pre-Calculus exam. This guide is designed to be quick, clear, and exactly what you need for a last-minute review. Let's dive in!

#

4.13 Matrices as Functions: The Big Picture

# Connecting Linear Transformations and Matrices

• Linear Transformation: Think of it as a function that moves and stretches vectors in a specific way. It keeps lines straight and the origin fixed.

• Transformation Matrix: This is a special matrix that represents a linear transformation. It's like a code that tells you how to transform any vector. For a 2D vector <x, y>, the transformation matrix is [a₁₁ a₁₂; a₂₁ a₂₂], which transforms the vector to <a₁₁x + a₁₂y, a₂₁x + a₂₂y>. 📜

  

  Image Created by Jed Q

• Image of a Vector: When you multiply a vector by a transformation matrix, the result is the transformed vector, also known as the image. 🔦

• Generalization: This concept extends to n-dimensional space. An n x n matrix represents a linear transformation in n-dimensional space. Each element aᵢⱼ is the coefficient for the xⱼ component of the output vector.

• Unit Vectors and Standard Basis: The unit vectors <1, 0> and <0, 1> (in 2D) are called the standard basis vectors. The transformation of these vectors tells you what the transformation matrix looks like. The transformed unit vectors become the columns of the matrix. 🗺️

  

  Image Courtesy of Towards Data Science

# Involving Angles: Rotation Matrices

• Rotation Matrix: A matrix that rotates a vector counterclockwise by an angle θ is given by [cos(θ) -sin(θ); sin(θ) cos(θ)]. 🌀

  

  Image Created by Jed Q

• Transformation Equations: When you multiply a vector <x, y> by the rotation matrix, the new vector <x', y'> is given by:

  $$x' = xcos(θ) − ysin(θ)$$

  $$y' = xsin(θ) + ycos(θ)$$

  This means the original vector is rotated by an angle θ counterclockwise. 📐

• Clockwise Rotation: To rotate clockwise by an angle θ, use the matrix [cos(-θ) -sin(-θ); sin(-θ) cos(-θ)]. ⏰

  

  Image Courtesy of Math Stack Exchange

# Absolute Values & Determinants

• Determinant: For a 2x2 matrix [a₁₁ a₁₂; a₂₁ a₂₂], the determinant is a₁₁a₂₂ - a₁₂a₂₁. ⏸
• Dilation Magnitude: The absolute value of the determinant tells you the magnitude of the dilation (scaling) of areas in R² under the transformation. 💡
  • Positive Determinant: Indicates a dilation (expansion). ➕
  • Negative Determinant: Indicates a reflection (flip) and a possible dilation. ➖

# Compositions of Two Linear Transformations

• Composition: When you apply one linear transformation after another, that's a composition. The output of the first transformation becomes the input of the second. 👗

• Associativity: The order of composition doesn't matter when you are composing 3 or more transformations, i.e. f(g(h(x))) = (f∘g)h(x) = f(g∘h(x)). 🙅

  

  Image Courtesy of Ximera (Ohio State University)

• Matrix Multiplication: The matrix of a composition is the product of the individual transformation matrices. If matrix A represents transformation f and matrix B represents transformation g, then matrix AB represents the composition g(f(x)). ⚡️

# Inverses of Linear Transformations

• Inverse Transformation: A transformation that "undoes" the effect of another transformation. If f maps x to y, then f⁻¹ maps y back to x. 🔃

• Formal Definition: If f: V → W and g: W → V are linear transformations, they are inverses if and only if g(f(x)) = x for all x in V and f(g(y)) = y for all y in W.

• Commutativity: The composition of a transformation and its inverse is commutative and results in the identity transformation, i.e. f(f⁻¹(x)) = f⁻¹(f(x)) = x. 🙋

• Inverse Matrix: If a linear transformation L is given by L(v) = Av, then its inverse L⁻¹ is given by L⁻¹(v) = A⁻¹v, where A⁻¹ is the inverse of matrix A. 🤝

• Identity Matrix: The inverse of matrix A, denoted as A⁻¹, satisfies A⁻¹A = I, where I is the identity matrix. Not every matrix has an inverse. A matrix is invertible if and only if its determinant is non-zero. 🤓

  

  Image Courtesy of Ximera (Ohio State University)

#Final Exam Focus

• High-Priority Topics:
  • Constructing transformation matrices from the transformation of unit vectors.
  • Rotation matrices and their applications.
  • Calculating determinants and understanding their geometric interpretation.
  • Composing linear transformations and multiplying their matrices.
  • Finding inverse transformations and matrices.
• Common Question Types:
  • Finding the matrix that represents a given transformation.
  • Rotating or reflecting vectors using matrices.
  • Finding the area of a transformed region using determinants.
  • Combining multiple transformations.
  • Determining if a matrix has an inverse and finding it.
• Last-Minute Tips:
  • Time Management: Quickly identify the type of transformation and apply the appropriate matrix.
  • Common Pitfalls: Pay close attention to the order of matrix multiplication in compositions and the sign of determinants.
  • Strategies: If you are stuck, try visualizing the transformation and how it affects the unit vectors.

#Practice Questions

Practice Question

Multiple Choice Questions

1. A linear transformation T in R² maps the vector <1, 0> to <2, -1> and the vector <0, 1> to <3, 4>. What is the matrix representation of T? (A) [2 3; -1 4] (B) [2 -1; 3 4] (C) [3 2; 4 -1] (D) [-1 2; 4 3]

2. A vector <5, -2> is rotated counterclockwise by 90 degrees. What is the resulting vector? (A) <2, 5> (B) <-2, -5> (C) <-2, 5> (D) <2, -5>

3. The determinant of the matrix [4 2; 1 3] is: (A) 10 (B) 14 (C) 16 (D) 8

Free Response Question

A linear transformation T is defined by a rotation of 60 degrees counterclockwise, followed by a reflection over the y-axis.

(a) Find the matrix that represents the rotation by 60 degrees counterclockwise.

(b) Find the matrix that represents the reflection over the y-axis.

(c) Find the matrix that represents the composite transformation T.

(d) Apply the composite transformation T to the vector <1, 2>.

Scoring Breakdown

(a) 2 points * 1 point for correctly identifying the rotation matrix structure * 1 point for correctly substituting 60 degrees into the rotation matrix

(b) 2 points * 1 point for correctly identifying the structure of the reflection matrix over the y-axis * 1 point for correctly filling in the values of the reflection matrix

(c) 3 points * 1 point for setting up the correct order of matrix multiplication * 2 points for correctly multiplying the matrices

(d) 2 points * 1 point for setting up the matrix-vector multiplication * 1 point for correctly multiplying the matrix and the vector

Answers:

Multiple Choice:

1. (A)
2. (A)
3. (A)

Free Response:

(a) The rotation matrix is [cos(60) -sin(60); sin(60) cos(60)] = [1/2 -sqrt(3)/2; sqrt(3)/2 1/2].

(b) The reflection matrix over the y-axis is [-1 0; 0 1].

(c) The composite transformation matrix is [-1 0; 0 1] * [1/2 -sqrt(3)/2; sqrt(3)/2 1/2] = [-1/2 sqrt(3)/2; sqrt(3)/2 1/2]

(d) Applying the composite transformation to the vector <1, 2> gives [-1/2 sqrt(3)/2; sqrt(3)/2 1/2] * [1; 2] = [-1/2 + sqrt(3), sqrt(3)/2 + 1]

You've got this! Remember to stay calm, take your time, and trust your preparation. Good luck on your exam! 🍀