Key takeaways
- Nine units, from special matrices to quadratic forms.
- A one-page list of eigenvalue properties, which the notes mark as very important for GATE.
- Solved examples on rank, consistency of equations and eigenvalues.
- Ten practice questions at the end for self-testing.
- Original Uncodemy revision notes following the common syllabus; check your own university syllabus.
About these handwritten notes
These are original Uncodemy revision notes for Linear Algebra and Matrices, written in a clean handwritten style so they feel like a topper's notebook rather than a textbook. They are short on purpose: each unit gives the key definitions, formulas, a solved example or two and the points that are easy to forget.
The notes follow the common syllabus that most Indian B.Tech universities use for this subject, including universities that follow the AICTE model curriculum. They are not official notes from any university. Unit names, order and depth differ from one university to another, so check your own university syllabus and mark any topic that is missing before you rely on the notes for an exam. The PDF is built for revision, so it keeps proofs short and focuses on the results and methods you use in numericals.
What's inside the linear algebra notes
The notes are arranged in nine units. One line on each:
Unit 1: Matrices, the basics
Order, transpose and matrix product, plus a table of special matrices (symmetric, skew-symmetric, orthogonal, Hermitian, unitary, idempotent, nilpotent, involutory) with their eigenvalue facts.
Unit 2: Determinants and inverse
Properties of determinants, singular matrices and the inverse using the adjoint, with a 2 x 2 check example.
Unit 3: Rank of a matrix
Rank by echelon form and normal form, with a solved example and shortcuts such as the rank of a column times a row being 1.
Unit 4: Systems of linear equations
The consistency test using the rank of A and of the augmented matrix, then Gauss elimination, Gauss-Jordan, Cramer's rule and LU decomposition, with a solved example.
Unit 5: Vector spaces
Linear dependence and independence, basis and dimension, and the four fundamental subspaces of an m x n matrix of rank r.
Unit 6: Linear transformations
Definition of a linear map, kernel and range, the dimension relation dim ker T + dim range T = dim V, and the 2D rotation matrix.
Unit 7: Eigenvalues and eigenvectors
The characteristic equation, a list of eigenvalue properties (trace, determinant, powers, inverse), a solved example and the Cayley-Hamilton theorem.
Unit 8: Diagonalisation and quadratic forms
P⁻¹AP = D, algebraic and geometric multiplicity, using diagonalisation to find powers, and the nature of a quadratic form from its eigenvalues.
Unit 9: Most asked questions (practice)
Ten mixed practice questions covering rank, consistency, eigenvalues and quadratic forms.
How to revise linear algebra with these notes
- Make row operations automatic. Rank, consistency and inverse all depend on clean row reduction. Practise until you can reduce a 3 x 3 or 3 x 4 matrix without errors.
- Learn the eigenvalue properties as a list. Sum of eigenvalues = trace, product = determinant, eigenvalues of A^k are λ^k, and so on. The notes point out that many GATE questions can be answered in about a minute with these, without full calculation.
- Solve, then check. After finding eigenvalues by hand, confirm them with the eigenvalue and eigenvector calculator. Check rank with the matrix rank calculator.
- Use the positive definite shortcut. The notes give a quick test: all leading principal minors greater than zero. Practise it on a few matrices.
- Write Cayley-Hamilton questions fully. Universities give marks for each step: characteristic equation, verification, and then A⁻¹ or a higher power.
- Connect the ideas. Rank links to consistency, consistency links to the null space, and the null space links to eigenvectors. Revising them together helps you remember.
Exam focus: high-weightage linear algebra topics
According to the tip at the end of the notes, GATE leans on eigenvalue properties, rank and consistency. University papers add longer questions on Cayley-Hamilton, diagonalisation and quadratic forms.
Topics worth extra time:
- Rank of a 3 x 4 matrix by echelon form.
- Consistency of a system of three equations, including "for what value of k does the system have a non-trivial solution?"
- Eigenvalues and eigenvectors of a 3 x 3 matrix, including repeated eigenvalues.
- Cayley-Hamilton: verify it and use it to find A⁻¹.
- Diagonalisation of a symmetric matrix and finding A⁴ from it.
- Linear independence of a set of vectors.
- Quadratic forms: reduce to canonical form and state its nature.
- Rank-nullity theorem: state it and verify it for a matrix.
Typical questions
The practice set includes questions such as "Test consistency and solve x + y + z = 6, x + 2y + 3z = 14, x + 4y + 9z = 36" and "Prove that the eigenvalues of a real symmetric matrix are real". Use them for practice; they are not a prediction of your paper.
Who these notes are for
- First-year B.Tech students studying the matrices unit of Engineering Mathematics 1, or a separate linear algebra paper.
- GATE aspirants revising linear algebra, which is part of the engineering mathematics section in most GATE papers.
- Data science and machine learning learners. Vectors, matrix products, eigenvalues and rank sit underneath PCA, regression and neural networks, so a solid base here makes those topics easier.
For unit-wise exam practice with answer outlines, pair these notes with the Engineering Mathematics 1 important questions PDF.
Common mistakes to avoid in linear algebra
- Changing rank with a wrong row operation. Only swap rows, multiply a row by a non-zero number, or add a multiple of one row to another. Dividing by an expression that could be zero is a common trap in "for what value of k" questions.
- Forgetting the augmented matrix. Consistency needs the rank of A and the rank of [A | B]. Comparing only one of them gives a wrong conclusion.
- Taking a zero vector as an eigenvector. An eigenvector must be non-zero. If your working gives only the zero vector, recheck the eigenvalue.
- Assuming every matrix can be diagonalised. The notes explain that a matrix is diagonalisable only when the geometric multiplicity equals the algebraic multiplicity for every eigenvalue.
- Writing AB = BA. Matrix multiplication is not commutative, so keep the order in (AB)⁻¹ = B⁻¹A⁻¹ and (AB)^T = B^T A^T.
- Skipping the check. Multiply A by A⁻¹, or A by an eigenvector, to confirm your answer before moving on.
A simple 4-day revision plan
- Day 1: Units 1 to 3. Special matrices, inverse and rank.
- Day 2: Units 4 to 6. Consistency, vector spaces and rank-nullity.
- Day 3: Units 7 and 8. Eigenvalues, Cayley-Hamilton, diagonalisation and quadratic forms.
- Day 4: Solve the ten practice questions with a timer, then revise the eigenvalue property list once more.
Download Linear Algebra Handwritten Notes PDF
Linear Algebra Handwritten Notes PDF
The PDF carries the same content as this page in a clean, print-ready layout, so you can read it offline and keep it for revision.
6 pages · 224 KB · PDF · Updated 5 Nov 2026