Linear Algebra
“Master vector spaces, four fundamental subspaces, Gaussian elimination, eigenvalues, spectral theorem, SVD, and PCA for GATE DA.”
A comprehensive academic study notebook covering vector spaces, subspaces, linear independence, basis and dimension, linear transformations, Rank-Nullity theorem, systems of linear equations, Gaussian elimination, determinants, eigenvalues and eigenvectors, Cayley-Hamilton theorem, diagonalizability, Gram-Schmidt orthogonalization, QR decomposition, symmetric matrices, Spectral Theorem, Positive Definiteness, Singular Value Decomposition (SVD), and Principal Component Analysis (PCA). Modeled on the reference Operating Systems benchmark for the official GATE Data Science & AI (DA) curriculum.
Structured Notes & Exam Lens
A · B.Tech Structured Study Sequence
5 Sections · Full CurriculumVector Spaces, Subspaces & Linear Transformations
Vector space axioms, subspace tests, linear span, bases, dimension, the four fundamental subspaces, the Rank-Nullity theorem, and matrix representations of linear maps.
Vector axioms, subspace closure criteria, spanning sets, linear independence, and basis dimensions.
Column space, null space, row space, left null space, orthogonal complements, and dim(V) = rank + nullity.
Linearity tests, standard matrix representations [T], kernel, range, and similarity transformations A' = P^{-1} A P.
Systems of Linear Equations & Determinants
Gaussian elimination, Row Echelon Forms, Rouché-Capelli consistency theorem, parametric solvability, and axiomatic determinant algebra.
Augmented matrices, REF/RREF, Rouché-Capelli theorem, unique solutions, infinite solutions, and inconsistency.
Axiomatic multilinearity, geometric volume interpretation, determinant identities, and adjugate inverses.
Eigenvalues, Diagonalization & Cayley-Hamilton
Characteristic equations, trace/determinant relations, Algebraic vs Geometric multiplicity, modal diagonalization, and Cayley-Hamilton matrix polynomial algebra.
Av = lambda v, characteristic polynomial det(A - lambda I) = 0, trace sum, and determinant product rules.
AM vs GM, modal matrix P, similarity diagonalization A = P D P^{-1}, and computing matrix powers A^k.
p(A) = 0, polynomial reduction of high powers, matrix inverse derivation, and the minimal polynomial monic divisor.
Orthogonality, QR Factorization & Spectral Theorem
Inner product geometry, Cauchy-Schwarz, Gram-Schmidt orthogonalization, QR decomposition, least-squares projections, real symmetric matrices, and positive definiteness.
Orthogonal bases, Gram-Schmidt process, QR decomposition A = QR, and least-squares projection matrices.
Real eigenvalues, orthogonal diagonalization A = Q Lambda Q^T, quadratic forms x^T A x, and Sylvester's criterion.
Singular Value Decomposition (SVD) & PCA
General matrix factorizations, singular values, left/right singular vectors, Eckart-Young theorem, covariance matrix diagonalization, and Principal Component Analysis.
A = U Sigma V^T, singular values sigma_i = sqrt(lambda_i), Eckart-Young rank-k truncation, and PCA dimensionality reduction.
B · GATE 2027 Syllabus Scope
Official syllabus scope and related lesson coverage; topics outside the syllabus are marked as enrichment.
- •Vector spaces, subspaces, linear dependence and independence of vectors, basis, dimension
- •Linear transformations, matrix representation of linear transformations, rank-nullity theorem
- •Systems of linear equations, Gaussian elimination, row echelon form, consistency and solutions
- •Determinants, axiomatic properties, Cramer's rule, matrix inverse via adjugate
- •Eigenvalues, eigenvectors, characteristic polynomial, trace and determinant relations
- •Cayley-Hamilton theorem, minimal polynomial, matrix powers and inverse calculations
- •Diagonalizability, algebraic multiplicity (AM) vs geometric multiplicity (GM), defective matrices
- •Inner product spaces, Cauchy-Schwarz inequality, Gram-Schmidt orthogonalization, QR decomposition
- •Projections, least-squares normal equations, orthogonal projection matrices
- •Symmetric matrices, Spectral Theorem, quadratic forms, positive definite and positive semi-definite matrices
- •Singular Value Decomposition (SVD), low-rank approximation, Principal Component Analysis (PCA)
Interactive Labs & Verified PYQs
Revision & Rapid Review Sheets
Authoritative Textbooks & Video Lectures
All notes in this notebook cross-reference standard academic literature. We do not pretend student notes substitute for classical texts — they guide you through them.
Curated lectures from verified educators (Gate Smashers, Neso Academy, Knowledge Gate, and MIT OCW) selected specifically for difficult concept visualizations.
These notes are open and maintained together.
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