#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>...