Linear Algebra Course Online: Pick One That Matches Your Exam

The right course is the one whose syllabus matches the exam's. It proves vector spaces and linear maps properly, then reaches eigenvalues, canonical forms, inner product spaces and quadratic forms. Rating, length and certificate matter far less. If you are learning linear algebra for machine learning, a general course is fine and this page is not aimed at you.

Both syllabi name every topic in the list, so in our view a course that skips one leaves you to teach yourself that topic from a textbook. The CSIR NET common syllabus for Parts B and C places linear algebra in Unit 1 beside real analysis. Our guide to the CSIR NET Mathematical Sciences syllabus covers the other units. If you are preparing for IIT JAM (the Joint Admission Test for MSc), compare the GATE and IIT JAM Mathematics syllabi first, because the table below covers only CSIR NET and GATE MA.

Both syllabi expect the full chain from vector spaces to canonical forms and quadratic forms. GATE MA adds Hermitian, normal and unitary matrices and numerical methods for linear systems. The topic names come from the CSIR-UGC NET Mathematical Sciences syllabus and the GATE 2026 MA syllabus.

The third column compares them with the four weekly modules of the DeepLearning.AI course Linear Algebra for Machine Learning and Data Science on Coursera (8, 9, 10 and 8 hours, 35 hours in total).

Our count, as of 14 October 2026: we split the linear algebra paragraphs of the two syllabi into these 10 rows and marked each against the course's four weekly outlines. One row is named fully, five in part and four not at all. "Partly" means the outline names only a piece of the row. This is one course, not every course, so run the same row-by-row check on the week-by-week syllabus of any course you are considering, ours included.

Plain-language glossary for the table and the check below:

• Rank: the dimension of the space a matrix maps onto.
• Kernel (null space): the vectors a matrix or map sends to zero.
• Nullity: the dimension of the kernel.
• Eigenvector and eigenvalue: a nonzero vector v and a number λ with Av = λv.
• Cayley–Hamilton theorem: every square matrix satisfies its own characteristic polynomial.
• Minimal polynomial: the lowest-degree polynomial p, with leading coefficient 1, for which p(A) = 0.
• Diagonalisable: similar to a diagonal matrix.
• Conjugate transpose (A*): the transpose with every entry replaced by its complex conjugate.
• Skew-symmetric: equal to minus its transpose.
• Hermitian: equal to its conjugate transpose (skew-Hermitian: equal to minus it).
• Normal: commutes with its conjugate transpose.
• Orthogonal: a real matrix whose transpose is its inverse.
• Unitary: a complex matrix whose conjugate transpose is its inverse.
• Gram–Schmidt process: turns a basis into an orthonormal one (mutually perpendicular unit vectors).
• Signature of a quadratic form: the number of positive and negative signs once the form is diagonalised.
• LU decomposition: writing a matrix as lower-triangular times upper-triangular.
• Jacobi and Gauss–Seidel iteration: step-by-step approximation methods for linear systems.

Popular courses are built for computation and intuition, while the exams reward knowing which structural fact applies to a given matrix. The DeepLearning.AI course spends its 35 hours on elimination, linear transformations, determinants, eigenvectors and principal component analysis (PCA), a data-reduction method. It asks for basic Python. That is the right design for machine learning work.

Three habits then mislead exam candidates:

• Treating a visual series as a course. Videos such as 3Blue1Brown's Essence of Linear Algebra build a picture of transformations and eigenvectors. They do not train you to find a null space or reduce a quadratic form. Use them for intuition, then work problems.
• Meeting the abstract jump unprepared. After matrices, a vector space is defined by rules, not by columns of numbers. Questions then treat polynomials, matrices and functions as vectors. Students who learned only coordinates feel lost at this point. Test every new definition on three examples: R^n, polynomials of degree at most n, and n by n matrices.
• Choosing a book by reputation. Strang is strong on computation and intuition. Hoffman and Kunze is rigorous and reaches canonical forms. Axler works through linear maps and postpones determinants. We recommend one rigorous main text plus a source of exam problems. Our guide to CSIR NET maths books names a main book per topic.

Answer the five questions below without notes, one point for each pass. These are our own questions, written from the syllabus topics above. Our recommendation for your score:

• 5: skip the course and use previous papers and mock tests.
• 3 or 4: fill the failed topics from one rigorous textbook, or from a free course whose weekly syllabus names those topics if you study well alone.
• 0 to 2: a structured course will save you time.

Can you find the dimension of a kernel without computing anything?

A linear map T goes from R^5 onto R^3. What is the dimension of its kernel?

• Pass: 2, by the rank–nullity theorem (dimension of the domain equals rank plus nullity, so 5 = 3 + nullity).
• Fail: any answer other than 2, or no answer. Writing down a matrix is not needed: rank–nullity gives the answer.

Can you decide diagonalisability from the minimal polynomial?

Take the matrix with rows (2, 1, 0), (0, 2, 0) and (0, 0, 3). Is it diagonalisable?

• Pass: No. The minimal polynomial is (x − 2)²(x − 3), which has a repeated root. Equivalently, the eigenspace for 2 has dimension 1 though 2 appears twice as an eigenvalue.
• Fail: answering yes because the eigenvalues are real or because there are three of them.

Can you list the Jordan forms of a nilpotent 4×4 matrix?

A nilpotent matrix has some power equal to zero. How many Jordan forms can a nilpotent 4×4 matrix have?

• Pass: 5, with block sizes (4), (3, 1), (2, 2), (2, 1, 1) and (1, 1, 1, 1). The last is the zero matrix.
• Fail: any other count, or a list that repeats a form.

Can you classify a quadratic form by signature?

Classify q(x, y) = x² + 4xy + y².

• Pass: Write it as (x + 2y)² − 3y², or use the symmetric matrix with rows (1, 2) and (2, 1), whose eigenvalues are 3 and −1. The form is indefinite, with rank 2 and one positive and one negative sign, so signature (1, 1).
• Fail: calling it positive definite, or being unable to start.

Can you say when a matrix is normal and what that buys you?

Is the matrix with rows (0, 1) and (−1, 0) normal? Can it be diagonalised?

• Pass: Normal means AA* = A*A. Here AAᵀ = I = AᵀA, so it is normal. Its eigenvalues are i and −i, so a unitary matrix diagonalises it over the complex numbers, but it is not diagonalisable over the real numbers.
• Fail: saying it cannot be diagonalised at all, or not knowing what normal means.

For a longer diagnosis of where algebra marks leak, use our CSIR-NET algebra weakness diagnostic.

Take a free NPTEL course only if you scored 3 or 4 on the check and study well alone. A score of 0 to 2 points to a structured batch, and a score of 5 needs neither. This is our recommendation.

NPTEL (the National Programme on Technology Enhanced Learning) ran a 12-week Linear Algebra course from January to April 2026 that was free to enrol in, with an optional paid certificate exam. As of 14 October 2026 that run has closed. Check the NPTEL course listing for the next run's enrolment dates, and compare its weekly syllabus with the ten rows above before you commit.

As of 14 October 2026, our recorded Linear Algebra course costs ₹2,799 (listed at ₹2,999). SB Sir teaches it in Hindi and English, from systems of equations and vector spaces through eigenvalues, diagonalisation, Jordan canonical form, quadratic forms, inner product spaces and invariant subspaces, with 481 pages of notes and CSIR NET previous-year questions.

If you also need real analysis and complex analysis, our Pure Mathematics combo (December 2026 live batch, taught in Hindi) costs ₹5,999 (listed at ₹7,999) for 180 days of access. It covers all three subjects in 250+ hours of live and recorded lectures. The linear algebra part runs through vector spaces, bases, linear transformations, rank–nullity, inner product spaces, eigenvalues, diagonalisation, Jordan canonical form and quadratic forms.

The combo adds daily practice sheets and advanced assignments, topic-wise previous-year question discussion for CSIR NET and GATE, and weekly tests with solution sessions. SB Sir teaches our Pure Mathematics courses. He is CSIR NET qualified with All India Rank 11 (104.5 marks, December 2017) and GATE MA qualified.

We publish this page and teach in this market, so our courses are among the options described here. Use the ten-row table to test ours as you would any other.

Our CSIR NET and GATE Mathematics 2027 course includes linear algebra. As of 14 October 2026 it costs ₹15,999 in one payment, or four monthly instalments of ₹3,999.75 (4 × 3,999.75 = 15,999), with access through 30 June 2027. It includes live classes, recorded lectures, assignments, previous-year question discussion, mock tests, weekly doubt sessions, personal mentoring and PDF notes.

Our digital courses carry a no-refund policy; a serious health issue immediately after purchase can be considered. So use our free mathematics study content (demo classes, PDF Drive resources, notes and previous-year questions) and the ₹999 demo plan, which gives 10 days of access to the 2027 course, before paying in full. Support runs 9am to 7pm, Monday to Saturday, and contact-form enquiries receive a reply within 24 hours.

Tell us your five-question score and your exam, and we will say which course fits: contact SBTech Math.