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Linear Algebra

Free Linear Algebra study guide: 8 units with summaries, key ideas, flashcards and 196 practice questions with step-by-step solutions.

Lines up with: GRE Mathematics Subject Test (algebra portion).

Units

1. UNIT 01

   Systems of linear equations

   A linear system can have one solution, none, or infinitely many. Row reduction is the method that decides which and finds them.
2. UNIT 02

   Matrix algebra

   Matrix algebra lets you treat whole systems as single objects. Multiplication and inverses are the key operations.
3. UNIT 03

   Determinants

   The determinant is one number that tells you whether a matrix is invertible and how it scales area or volume.
4. UNIT 04

   Vector spaces & subspaces

   Vector spaces are the setting for all of linear algebra. Span, independence, basis and dimension describe their structure.
5. UNIT 05

   Linear transformations

   A linear transformation preserves addition and scaling. Every one between finite-dimensional spaces is a matrix.
6. UNIT 06

   Eigenvalues & eigenvectors

   Eigenvectors are directions a matrix only stretches. Eigenvalues are the stretch factors. They unlock diagonalization and long-run behavior.
7. UNIT 07

   Orthogonality & least squares

   Orthogonality brings geometry into linear algebra: lengths, angles, projections and the best approximate solution to systems with no exact answer.
8. UNIT 08

   Symmetric matrices & SVD

   Symmetric matrices have especially nice structure. The singular value decomposition extends that structure to every matrix.