Linear Algebra | Index

Linear algebra is the study of vectors, matrices and linear transformations, and of systems of linear equations of any size.

The course covers row reduction, vector equations, linear transformations, matrix algebra, determinants, vector spaces, eigenvalues and orthogonality, ending with least squares and the singular value decomposition.

It’s for college students in a first linear algebra course, usually taken after or alongside Calculus 2 or Calculus 3.

What to know first: matrix operations, matrix multiplication, determinants and inverse matrices in Precalculus, Unit 4, plus vectors and the dot product in Calculus 3, Unit 1. Calculus is used only in a few examples.

Lessons for this course are coming soon. The list below shows every planned lesson in teaching order.

Units in this course

Unit 1: Systems of Linear Equations

  1. Gaussian Elimination — find pivot positions, reduce to row echelon form and tell whether a system has no, one or infinitely many solutions
  2. Reduced Row Echelon Form — use Gauss-Jordan elimination to reach reduced row echelon form and read off the solution
  3. Applications of Linear Systems — balance chemical equations and analyze network flow

Also see: solving systems with augmented matrices in Precalculus, Unit 4, and systems of equations in three variables in Algebra 2, Unit 2.

Unit 2: Vectors and the Matrix Equation

  1. Linear Combinations — write a vector as a combination of other vectors and solve the vector equation
  2. Span of Vectors — decide whether a set of vectors spans a line, a plane or all of ℝⁿ
  3. Matrix Equation Ax = b — multiply a matrix by a vector and decide when Ax = b has a solution
  4. Homogeneous Systems — when Ax = 0 has only the trivial solution and when it has others
  5. Parametric Vector Form — write infinitely many solutions with free variables in parametric vector form
  6. Linear Independence — test whether a set of vectors is linearly independent or dependent

Also see: vectors in three dimensions in Calculus 3, Unit 1.

Unit 3: Linear Transformations

  1. Linear Transformations — check whether a function from ℝⁿ to ℝᵐ is linear
  2. Standard Matrix of a Linear Transformation — find the standard matrix from T(e₁), …, T(eₙ) for any map from ℝⁿ to ℝᵐ
  3. One-to-One and Onto Transformations — decide whether a linear transformation is one-to-one, onto or both
  4. Composition of Linear Transformations — combine transformations by multiplying their matrices

Also see: matrix transformations in the plane in Precalculus, Unit 4.

Unit 4: Matrix Algebra

  1. Transpose of a Matrix — transpose a matrix, use its rules and recognize symmetric matrices
  2. Elementary Matrices — write row operations as matrices and see why the [A | I] method works
  3. Invertible Matrix Theorem — the many equivalent ways to tell whether a square matrix is invertible
  4. LU Decomposition — factor a matrix (when possible) into lower and upper triangular matrices to solve systems faster

Also see: matrix operations, matrix multiplication and inverse matrices in Precalculus, Unit 4.

Unit 5: Determinants

  1. Cofactor Expansion — find determinants of 4 × 4 and larger matrices by expanding along the best row or column
  2. Properties of Determinants (coming soon) — see how row operations, products and transposes change a determinant
  3. Adjugate Matrix (coming soon) — find the inverse of a matrix with cofactors and the adjugate
  4. Determinants as Area and Volume (coming soon) — find the volume of a parallelepiped and how a linear transformation scales area and volume

Also see: 2 × 2 and 3 × 3 determinants and Cramer’s rule in Precalculus, Unit 4.

Unit 6: Vector Spaces

  1. Vector Spaces (coming soon) — the vector space axioms, with polynomials, matrices and functions as examples
  2. Subspaces (coming soon) — test whether a subset of a vector space is a subspace
  3. Null Space and Column Space (coming soon) — find and describe the null space and column space of a matrix
  4. Basis and Dimension (coming soon) — find a basis and the dimension of a vector space or subspace
  5. Rank of a Matrix (coming soon) — find the rank and dimension and use the rank-nullity theorem
  6. Change of Basis (coming soon) — write coordinate vectors relative to a basis and convert between bases

Unit 7: Eigenvalues and Eigenvectors

  1. Eigenvalues and Eigenvectors (coming soon) — what Av = λv means, how to check an eigenvector and how to find the eigenspace for an eigenvalue
  2. Characteristic Polynomial (coming soon) — find eigenvalues by solving det(A − λI) = 0, including 2 × 2 matrices with complex eigenvalues
  3. Diagonalization (coming soon) — write A = PDP⁻¹, use similar matrices and find powers of a matrix

Also see: eigenvalue methods for systems of differential equations in Differential Equations, Unit 4, and Markov chains in Finite Math and Business Calculus, Unit 4.

Unit 8: Orthogonality and Least Squares

  1. Orthogonal Sets and Orthonormal Bases (coming soon) — use orthogonal and orthonormal bases and orthogonal matrices
  2. Orthogonal Projections (coming soon) — project a vector onto a subspace and find its distance from the subspace
  3. Gram-Schmidt Process (coming soon) — turn any basis into an orthogonal or orthonormal basis
  4. QR Factorization (coming soon) — factor A = QR, where Q has orthonormal columns and R is upper triangular
  5. Least-Squares Solutions (coming soon) — find the best approximate solution of Ax = b with the normal equations
  6. Orthogonal Diagonalization (coming soon) — diagonalize a symmetric matrix with an orthogonal matrix (the spectral theorem)
  7. Singular Value Decomposition (coming soon) — factor any matrix as UΣVᵀ and see what the singular values measure

Also see: the dot product (Precalculus), vector projections (Calculus 3) and the least-squares regression line (Statistics).

For Teachers: Standards Alignment

Lessons follow a typical first college course in linear algebra, matrix-based and applied, as taken by math, science, engineering and computer science majors. There is no AP exam for this course. Matrix arithmetic, 2 × 2 and 3 × 3 determinants, inverse matrices and Cramer’s rule are taught in Precalculus, Unit 4, which matches the matrix topics in the Common Core high school standards (N-VM).

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