Linear algebra

Linear algebra is a field of mathematics involving vectors, matrices, and spaces. More broadly, it studies the behaviour of linear maps in $n$-dimensional space. It’s foundational to modern science and engineering theory and practice.

Key concepts

• Linear equation
  • Row reduction
  • Cramer’s rule
• Vector
  • Dot product
  • Cross product
  • Vector-valued function
  • Projection
• Matrices and linear transformations
  • Transpose
  • Matrix multiplication
  • Determinant
  • Matrix similarity
  • Types of matrices
    • Hessian matrix
    • Jacobian matrix
  • Inverse matrix
    • Moore-Penrose pseudoinverse
• Vector space and subspace
  • Image
  • Kernel subspace
  • Rank-nullity theorem
• Linear combination
  • Linear independence
    • Wronskian determinant
  • Basis
    • Change of basis
• Orthogonality
  • Gram-Schmidt algorithm
• Eigenvalue and eigenvector and diagonalisation
  • Matrix trace
  • Markov chain
• Least squares
• Matrix decomposition
  • Eigendecomposition
  • Singular value decomposition
  • Spectral theorem
  • LU decomposition
  • QR decomposition
• Tensor

Resources

For introductory resources:

• Linear Algebra with Applications, by Otto Bretscher
• Introduction to Linear Algebra, by Gilbert Strang

For a more rigorous introduction:

• A First Course in Random Matrix Theory, by Marc Potters and Jean-Philippe Bouchard
• Linear Algebra Done Right, by Sheldon Axler

See also

• MAT188 — Linear Algebra, from first-year of undergrad