∑GATE DA Atlas
GATE DA / Section 2

Linear Algebra

Vector spaces and subspaces, linear independence, bases and dimension, special matrices (projection, orthogonal, idempotent, partitioned), systems of linear equations and Gaussian elimination, rank and nullity, determinants, eigenvalues and eigenvectors, projections and quadratic forms, and the LU and singular value decompositions.

8 topics75 practice questions≈ 10–17 marksPractise this section →

Linear algebra is the language of data science: a data set is a matrix, a model's parameters are a vector, PCA is an eigendecomposition, least squares is a projection, and neural networks are stacks of matrix multiplications. Third-party analyses put it at roughly 10 marks in GATE DA 2024 and 17 marks in 2025.

What the syllabus lists: vector space, subspaces, linear dependence and independence of vectors, matrices, projection matrix, orthogonal matrix, idempotent matrix, partition matrix and their properties, quadratic forms, systems of linear equations and solutions, Gaussian elimination, eigenvalues and eigenvectors, determinant, rank, nullity, projections, LU decomposition, singular value decomposition.

Topics

Read in order — later topics build on earlier ones
01Vector Spaces, Subspaces, Independence, Basis and DimensionThe axioms of a vector space with the standard examples, the subspace test, span and linear combinations, linear dependence and independence and how to test it, bases, dimension and coordinates, sums and intersections of subspaces, and inner products, orthogonality and Gram–Schmidt.0/9 solved02Matrices — Operations, Special Matrices and Partitioned MatricesMatrix algebra and its rules (transpose, inverse, trace), symmetric and skew-symmetric matrices, orthogonal matrices, idempotent and projection matrices, nilpotent and involutory matrices, permutation matrices, and partitioned (block) matrices with block multiplication, block determinants and the Schur complement.0/10 solved03Systems of Linear Equations and Gaussian EliminationWriting systems as Ax = b, elementary row operations, row echelon and reduced row echelon forms, Gaussian and Gauss–Jordan elimination with back substitution, pivoting and cost, the rank test for consistency (none, unique or infinitely many solutions), homogeneous systems, parametric solutions, Cramer's rule and computing inverses.0/9 solved04Rank, Nullity and the Four Fundamental SubspacesColumn space, row space, null space and left null space, the definition and computation of rank, the rank–nullity theorem, rank inequalities for sums and products, rank of special matrices (outer products, Gram matrices), full-rank matrices and the invertible matrix theorem.0/9 solved05DeterminantsDeterminants by cofactor expansion and by elimination, the effect of row operations, multiplicativity and the other algebraic rules, the adjugate and inverse formula, geometric meaning as signed volume, and special determinants (triangular, block, Vandermonde) — with the shortcuts GATE expects.0/9 solved06Eigenvalues and EigenvectorsThe eigenvalue equation and characteristic polynomial, eigenspaces, algebraic and geometric multiplicity, trace and determinant shortcuts, eigenvalues of powers, inverses, shifts and special matrices, diagonalisation, the spectral theorem for symmetric matrices, the Cayley–Hamilton theorem, and why eigenvectors matter in PCA, Markov chains and the power method.0/10 solved07Projections, Least Squares and Quadratic FormsOrthogonal projection onto a line and onto a subspace, projection matrices and their properties, least squares and the normal equations, distance to a subspace, quadratic forms and their symmetric matrices, definiteness tests (eigenvalues, leading minors), the Rayleigh quotient, and diagonalising a quadratic form by a change of variables.0/9 solved08LU Decomposition and Singular Value DecompositionLU factorisation as recorded Gaussian elimination, permutations (PA = LU), solving systems with forward and back substitution, LDU and Cholesky; the singular value decomposition, how to compute it from AᵀA, its geometric meaning, rank, matrix norms and condition number, the Eckart–Young low-rank approximation, the pseudo-inverse, and the link to PCA.0/10 solved