Math Ladder · College

Linear Algebra

Systems and row reduction, vector spaces and linear maps, determinants, eigenvalues, orthogonality, least squares and the SVD.

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49 topics in 8 units. Builds on Calculus 2: Integrals and Series. Leads to Differential Equations, Numerical Analysis and Abstract Algebra. Not sure this is the right course? Find your rung.

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Unit 1 Linear Systems

6 topics

Solve systems with row reduction and describe all their solutions.

  1. Linear Systems and Their Pictures
  2. Augmented Matrices and Row Operations
  3. Gaussian Elimination and Echelon Form
  4. Reduced Row Echelon Form
  5. Free Variables and Parametric Solutions
  6. When Solutions Exist and Are Unique

Unit 2 Vectors and Matrix Equations

9 topics

Combine vectors, describe spans, and test independence.

  1. Vectors in Rⁿ and Linear Combinations
  2. Span
  3. The Matrix Equation Ax = b
  4. Homogeneous Systems
  5. Linear Independence
  6. Matrix Operations and Transposes
  7. Inverses by Gauss–Jordan
  8. The Invertible Matrix Theorem
  9. Elementary Matrices and LU Factorization

Unit 3 Linear Transformations

6 topics

Describe linear maps by matrices and see their geometry.

  1. Linear Transformations
  2. The Matrix of a Linear Transformation
  3. Rotations, Reflections, Shears and Projections
  4. Composition as Matrix Multiplication
  5. One-to-One and Onto
  6. Neural Network Layers: Matrices and Switches

Unit 4 Determinants

4 topics

Compute determinants and see them as volume scale factors.

  1. Determinants by Cofactor Expansion
  2. Properties of Determinants
  3. Determinants as Area and Volume
  4. Cramer's Rule and the Adjugate

Unit 5 Vector Spaces

7 topics

Generalize to vector spaces, subspaces, bases, dimension and change of basis.

  1. Writing Short Proofs
  2. Vector Spaces and Subspaces
  3. Null Space and Column Space
  4. Bases
  5. Dimension and Rank
  6. The Rank–Nullity Theorem and the Four Subspaces
  7. Coordinates and Change of Basis

Unit 6 Eigenvalues and Eigenvectors

7 topics

Find the directions a matrix only stretches, and diagonalize.

  1. Eigenvectors and Eigenvalues
  2. The Characteristic Equation
  3. Eigenspaces and Multiplicity
  4. Diagonalization
  5. Matrix Powers and Markov Chains
  6. Complex Eigenvalues
  7. Similar Matrices

Unit 7 Orthogonality and Least Squares

6 topics

Project onto subspaces, build orthonormal bases, and fit data by least squares.

  1. Length and Orthogonality in Rⁿ
  2. Orthogonal Complements
  3. Orthogonal Projections
  4. Gram–Schmidt and QR
  5. Least Squares
  6. Orthogonal Matrices

Unit 8 Symmetric Matrices and the SVD

4 topics

Diagonalize symmetric matrices, study quadratic forms, and use the SVD.

  1. Symmetric Matrices and the Spectral Theorem
  2. Quadratic Forms and Positive Definite Matrices
  3. The Singular Value Decomposition
  4. Low-Rank Approximation and Applications
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