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Linear Algebra

Linear Algebra is the language of machine learning and a core GATE DA subject. These lessons explain vector spaces, rank, special matrices, quadratic forms, eigenvalues and the LU and singular value decompositions, with small programs in plain Python (exact fractions where possible) that check every matrix and number shown.

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1Vector Spaces

  1. Vector Spaces, Subspaces and Linear IndependenceVectors and vector spaces, subspaces and how to test them, linear combinations and span, linear independence, basis and dimension, and the dot product, norms and orthogonality, for GATE DA.

2Matrices

  1. Matrices, Rank and Systems of Linear EquationsMatrix operations, row echelon form and Gaussian elimination, the rank of a matrix, column space and null space, the rank-nullity theorem, and when a system Ax = b has no solution, one solution or infinitely many, for GATE DA.
  2. Special Matrices - Symmetric, Orthogonal, Idempotent, Projection and PartitionedDeterminants and their properties, symmetric and skew-symmetric matrices, orthogonal matrices, idempotent and projection matrices including the hat matrix of regression, and partitioned (block) matrices, for GATE DA.
  3. Quadratic Forms and DefinitenessQuadratic forms xᵀAx and their symmetric matrices, positive and negative definite, semidefinite and indefinite matrices, the eigenvalue test and Sylvester's criterion, and why definiteness matters for optimization and covariance matrices, for GATE DA.

3Eigenvalues and Decompositions

  1. Eigenvalues, Eigenvectors and DiagonalizationEigenvalues and eigenvectors, the characteristic polynomial, trace and determinant, eigenvalues of special and transformed matrices, algebraic and geometric multiplicity, diagonalization, the Cayley-Hamilton theorem and the power method, for GATE DA.
  2. LU DecompositionWriting a matrix as a lower triangular times an upper triangular matrix, Doolittle's method from Gaussian elimination, solving Ax = b with forward and back substitution, when LU exists, pivoting (PA = LU), the determinant from LU, and Cholesky decomposition, for GATE DA.
  3. Singular Value Decomposition (SVD)The singular value decomposition A = UΣVᵀ, how it relates to the eigenvalues of AᵀA and AAᵀ, computing it by hand, rank and norms from singular values, low-rank approximation (Eckart-Young), the pseudo-inverse, and its use in PCA and least squares, for GATE DA.

4GATE Practice

  1. Linear Algebra GATE Practice SetMixed GATE DA style questions on vector spaces, rank, linear systems, special matrices, quadratic forms, eigenvalues, LU decomposition and SVD, with answers and explanations.