Introduction to Linear Algebra
Chapter 1: Foundations
What is a Vector?
Introduction to vectors as arrows and lists of numbers.
Requirements:
Basic Algebra
Difficulty:
1
Vector Addition
How to add vectors geometrically and algebraically.
Requirements:
Vectors
Difficulty:
1
Linear Combinations
The fundamental building block of linear algebra.
Requirements:
Vectors
Difficulty:
2
What is Span?
Understanding the space reachable by a set of vectors.
Requirements:
Linear Combinations
Difficulty:
2
Linear Independence
Why some vectors are "redundant" in a set.
Requirements:
Span
Difficulty:
2
Basis of a Vector Space
The set of vectors that perfectly describe a space.
Requirements:
Independence
Difficulty:
3
Dimension Explained
Why we call a line 1D and a plane 2D.
Requirements:
Basis
Difficulty:
2
Chapter 2: Matrices
Introduction to Matrices
Thinking of matrices as data structures.
Requirements:
Basic Algebra
Difficulty:
1
Matrix-Vector Product
The first step toward linear transformations.
Requirements:
Vector Operations
Difficulty:
2
Matrix Multiplication
How to combine two matrix transforms into one.
Requirements:
Matrix-Vector Product
Difficulty:
2
The Identity Matrix
The "Number 1" of the matrix world.
Requirements:
Matrix Mult.
Difficulty:
1
The Transpose
Flipping a matrix across its diagonal.
Requirements:
Intro to Matrices
Difficulty:
2
The Inverse Matrix
How to "undo" a linear transformation.
Requirements:
Determinants
Difficulty:
3
Chapter 3: Transformations
Linear Transformations
Viewing matrices as actions/functions on space.
Requirements:
Matrix-Vector Product
Difficulty:
3
Rotations and Scaling
Using matrices to spin and stretch 2D objects.
Requirements:
Linear Transf.
Difficulty:
3
Shear Transformations
Distorting space while keeping lines parallel.
Requirements:
Linear Transf.
Difficulty:
3
Composition of Transformations
Applying multiple matrices in sequence.
Requirements:
Matrix Mult.
Difficulty:
3
Determinants - The Area Factor
How matrices change the size of shapes.
Requirements:
Linear Transf.
Difficulty:
3
Affine Transformations
Adding translations to our toolkit
Requirements:
Linear Transf.
Difficulty:
3
Chapter 4: Linear Systems
Systems of Linear Equations
Intersecting lines and planes.
Requirements:
Basic Algebra
Difficulty:
1
Augmented Matrices
A shorthand for writing systems of equations.
Requirements:
Systems of Eq.
Difficulty:
1
Elementary Row Ops
The three legal moves in matrix algebra.
Requirements:
Intro to Matrices
Difficulty:
2
Gaussian Elimination
The step-by-step algorithm to solve systems.
Requirements:
Systems of Eq.
Difficulty:
2
Row Echelon Form (REF)
The goal of Gaussian elimination.
Requirements:
Gaussian Elim.
Difficulty:
2
Reduced Row Echelon (RREF)
Simplifying systems to find unique solutions.
Requirements:
REF
Difficulty:
2
Chapter 5: Vector Spaces
Rank of a Matrix
Counting the "useful" information in a matrix.
Requirements:
RREF
Difficulty:
3
Null Space (Kernel)
Finding vectors that a matrix squashes to zero.
Requirements:
Linear Transf.
Difficulty:
3
Column Space (Image)
The set of all possible outputs of a matrix.
Requirements:
Span
Difficulty:
3
Row Space
The space spanned by the rows of a matrix.
Requirements:
Column Space
Difficulty:
3