Linear Algebra, from matrices to the ideas behind them
Vector spaces, linear maps, eigenvalues and inner products — one-to-one and online, for first-year courses and the more abstract second course that follows.
What this covers in detail
Vector spaces & linear maps
Subspaces, span, linear independence, bases and dimension, kernel and image, rank–nullity and change of basis.
Eigenvalues & diagonalisation
Determinants, characteristic polynomial, eigenvectors, diagonalisation, Jordan form and matrix exponentials.
Inner products & orthogonality
Inner product spaces, Gram–Schmidt, orthogonal projections, least squares, the spectral theorem and the SVD.
Linear Algebra usually arrives as matrix arithmetic and then, a few weeks in, turns abstract: subspaces, bases, linear maps. That turn is where most students lose the thread. These lessons connect the two — every abstract idea tied back to a matrix you can compute with.
What this covers
Abstract ideas, concrete examples
Each definition comes with a matrix you can compute — so a basis or a kernel is something you have seen, not just read.
Computation and proof
The row reduction that the exam times you on, and the short proofs it asks you to write — both practised.
Your course, your notation
Lessons follow your lecture notes and problem sheets, whether your course is applied, pure or somewhere between.
Pricing
Choose the support that fits.
One-to-one tutoring, from a single session to consistent weekly progress.
Your first 30 minutes are free.
Start with a free 30-minute assessment. We will understand what the student needs and recommend the right plan.
Every plan includes live one-to-one lessons.
The Noktua practice platform is free with every account — even if you have not booked a lesson yet.
Lessons at the student's home carry an extra charge of €10/hour.
Common questions
Which courses is this for?
First- and second-year Linear Algebra in maths, engineering, physics, economics and computer science degrees, plus the linear algebra inside machine-learning courses.
My course is very applied. Is that covered?
Yes. For engineering and data courses the focus shifts to computation, least squares and the SVD; for pure courses to proofs. The assessment decides which.
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