Linear Algebra Tuition in Singapore

Linear Algebra

The most natural place to begin is where you already have some footing: vectors, matrices, and systems of equations. From there I help you see how the same ideas grow into vector spaces, modules, and the maps between them. It is a subject where careful proofs and hands-on computation meet, and it turns up everywhere in engineering, physics, computer science and machine learning.

Systems of linear equations, Gaussian elimination, matrices, determinants
Vector spaces, subspaces, span, linear independence, basis and dimension
Linear transformations, homomorphisms, rank-nullity theorems, isomorphism theorems, change of basis, matrix representations, dual spaces
Eigenvalues and eigenvectors, diagonalisation, Jordan-Canonical Form, rational canonical form
Inner products, orthogonality, the Gram-Schmidt process, normal, self-adjoint, unitary operators, the method of least squares
Singular Value Decomposition, Polar Decomposition, bilinear forms, quadratic forms

Where linear algebra usually trips students up

Linear algebra rewards practice, yet the marks that decide a grade tend to slip away in the same few places.

Computing without understanding. You can row-reduce a matrix perfectly and still struggle to say what a basis or a rank actually is, and that gap is where the harder marks live.
The jump into abstraction. Moving from arrows in the plane to vector spaces of polynomials or functions, where a vector no longer looks like a list of numbers, unsettles almost everyone at first.
Span, independence and basis blurring together. The three ideas are closely related but not the same, and keeping them distinct is half the battle in the early proofs.
Confusing a linear map with its matrix. The same transformation has different matrices in different bases, and losing sight of that makes change of basis and diagonalisation feel arbitrary.
Treating eigen-theory as mechanics only. Finding eigenvalues is routine; knowing why we diagonalise, and what to do when a matrix will not, is what the exam really tests.
Hand-waving the short proofs. Much of the assessment is brief structural arguments, and marks are lost to vagueness rather than to hard computation.

Before you start: a linear algebra prerequisites checklist

Linear algebra leans on prior calculus far less than most university maths, but a few foundations make the abstract parts land much sooner. Before week one, it pays to make sure the following feel solid.

What to be comfortable with before week one
Be comfortable with	Why it matters for linear algebra
Algebra and arithmetic	Confident manipulation of equations and fractions keeps the row-reduction bookkeeping from getting in the way of the ideas.
Vectors and geometry	A feel for vectors, lines and planes in two and three dimensions anchors the abstract definitions that follow.
Functions and maps	Linear transformations are functions, so domain, range and composition should already be familiar.
Summation notation	Sums over indices appear everywhere, from matrix products to inner products.
Complex numbers	A little comfort here helps, since a real matrix can have complex eigenvalues.
Proof and logic	Some exposure to writing simple proofs makes the move from computation to justification much smoother.

A one-semester self-study plan for independent learners

Linear algebra is one of the most rewarding subjects to teach yourself, because the computations give quick feedback while the theory rewards patience. If you are working through it alone, perhaps for a computing, data-science or engineering major, here is a thirteen-week route map. Set the pace to suit your background, and get in touch if a particular idea, often change of basis or the spectral theorem, refuses to settle.

Texts I often point self-study students to
Text	Author	Best for
Introduction to Linear Algebra	Gilbert Strang	Intuition- and applications-led; the most popular first course
Linear Algebra Done Right	Sheldon Axler	A clean, proof-first treatment that builds the theory without leaning on determinants
Linear Algebra	Friedberg, Insel & Spence	A rigorous, well-balanced standard course text
Linear Algebra	Hoffman & Kunze	A classic for the more advanced material, including canonical forms
Matrix Analysis	Horn & Johnson	A reference for the matrix-theoretic results that applied courses lean on

A thirteen-week plan
Week	Focus	Suggested reading	Checkpoint
1–2	Systems of equations and matrices	Strang or Friedberg, the opening chapters	Solve any linear system by Gaussian elimination and read off its structure
3	Determinants	Strang or Friedberg	Compute determinants and know what they measure
4	Vector spaces and subspaces	Axler or Friedberg	Recognise vector spaces and subspaces beyond Rⁿ
5	Span, independence, basis, dimension	Axler or Friedberg	Decide independence and produce a basis with confidence
6	Linear transformations and matrices	Axler or Friedberg	Represent a map by a matrix and recover the map from the matrix
7	Rank-nullity and change of basis	Friedberg	Apply the rank-nullity theorem and change coordinates cleanly
8	Eigenvalues and eigenvectors	Strang or Friedberg	Find eigenvalues and eigenvectors and interpret them
9	Diagonalisation	Friedberg	Diagonalise where possible and recognise when it fails
10	Inner products and orthogonality	Axler or Friedberg	Use the Gram-Schmidt process and orthogonal projection
11	Least squares	Strang	Set up and solve a least-squares problem
12	The spectral theorem	Axler	Orthogonally diagonalise symmetric and normal operators
13	Singular value decomposition and review	Strang or Horn & Johnson	Form a singular value decomposition and explain what it reveals

Frequently Asked Questions (FAQ) - Linear Algebra Tuition

Which linear algebra topics and modules do you cover?

Everything from a first matrix-algebra course to honours material: systems of equations, vector spaces, linear maps, eigenvalues and diagonalisation, inner-product spaces, and the more advanced canonical forms and the singular value decomposition. That spans the first- and second-year linear algebra modules at NUS, NTU, SUTD, SIT and the other local universities. Send me your module outline and I will confirm what I would cover.

I can do the calculations but lose marks on proofs. Can you help?

Yes, and it is the most common reason linear-algebra students come to me. We work on stating definitions precisely, writing the short structural proofs the subject is built on, and reading a problem closely enough to see which definition it is really testing. The computation usually takes care of itself.

How relevant is linear algebra to machine learning and data science?

Central. Vectors, matrices, projections, eigenvalues and the singular value decomposition are the language those fields are written in. If that is your goal, I can keep the lessons anchored to the ideas that carry the most weight downstream, without skipping the understanding that makes them usable.

Do you cover honours topics like canonical forms and the spectral theorem?

Yes. Jordan and rational canonical forms, the spectral theorem for normal operators, bilinear and quadratic forms, and the singular value and polar decompositions are all part of the work I teach, and they sit on this same page rather than a separate one.

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