Linear algebra tutoring, online and with the right vocabulary.
Vector spaces, matrices, systems with a parameter, eigenvalues: linear algebra is the exam where vocabulary matters as much as calculation, and where one abstract idea never understood blocks everything that follows. We work 1-to-1 online, with concrete examples before definitions.
Let’s start from where you are stuck now.
Who asks us for help with Linear algebra.
- You study the definitions but cannot use them in exercises: basis, kernel, image, rank.
- You get the discussion of systems with a parameter wrong, or diagonalisation.
- You have a single Geometry and Linear Algebra exam and must hold together the abstract part and the part about figures.
- You are preparing the oral and must state definitions and theorems precisely.
How we work on Linear algebra.
- The example before the definition: every concept starts from a small case that can be written out in full, then it is generalised.
- Results checked in two ways: rank, kernel, dimension and systems are linked inside the exercises, so every answer has a cross-check.
- Timed practice with a practice test and a guided review of the mistakes, before the exam.
What Linear algebra covers.
Contents change from one course and one lecturer to another: this is the structure we meet most often. Your course syllabus has the last word: in the first lesson we read it together and work on that.
- Matrices and systems
- Operations on matrices, the inverse matrix
- Gaussian elimination
- Kronecker-Capelli and systems with a parameter
- Vector spaces
- Subspaces and linear combinations
- Linear dependence, bases and dimension
- Grassmann's formula
- Linear maps
- The associated matrix and changes of basis
- Kernel, image and the rank-nullity theorem
- Isomorphisms
- Determinant and rank
- The determinant and Laplace expansion
- Rank by minors and by row reduction
- Invertibility of a matrix
- Eigenvalues and diagonalisation
- Characteristic polynomial, eigenvalues and eigenvectors
- Algebraic and geometric multiplicity
- Symmetric matrices and the spectral theorem
- Inner product and quadratic forms
- Orthogonality and the Gram-Schmidt process
- Quadratic forms and the study of their sign
How the exam is usually structured.
The linear algebra written paper usually has four or five exercises: a system to discuss as a parameter varies, a matrix to diagonalise, a basis or a subspace to find, often a quadratic form or an inner product. The oral, where there is one, asks for definitions and theorems with the precision of a statement and sometimes a short proof. The format depends on the lecturer: the past papers of your course are the most reliable source.
Compared with calculus the calculation is short but the reasoning is long: a 3 by 3 matrix reduces in a few minutes, and the mark depends on what you conclude from it. That is why being able to say in your own words what each object means matters: if you do not know what a kernel is, you do not recognise it when it is needed.
The biggest risk is studying by formula. The results of linear algebra are connected to each other (rank, kernel, dimension, systems), and whoever keeps them separate loses the shortcut that lets you check a result in two different ways.
Three mistakes that cost marks.
- Confusing rank, kernel and image
The rank is the dimension of the image, and the dimension of the kernel is n minus the rank, where n is the number of columns. Swapping these numbers in the discussion of a system leads to «unique solution» where there are infinitely many. Before concluding, always write n, the rank of the coefficients and the rank of the augmented matrix.
- Declaring a matrix diagonalisable by looking only at the eigenvalues
Having all the eigenvalues in the field is not enough: for each eigenvalue the geometric multiplicity, the dimension of the eigenspace, must equal the algebraic one. It is the check that gets skipped, and an exercise with a parameter hides it in a single value.
- Using the determinant for everything
The determinant tells you whether a matrix is invertible, but it does not give the rank when it is zero and it becomes slow beyond 3 by 3. Row reduction gives rank, kernel and solutions in one go and can be checked row by row.
An exam-style exercise, step by step.
A diagonalisation with a parameter
For which values of is the matrix
diagonalisable?
The matrix is upper triangular, so the eigenvalues sit on the diagonal: with algebraic multiplicity and with algebraic multiplicity .
For there is nothing to check: the eigenspace has dimension . Everything is decided at , where we need an eigenspace of dimension . We compute
If the nonzero rows are and , independent: the rank is and the eigenspace has dimension . If only remains: rank and an eigenspace of dimension .
Conclusion: is diagonalisable if and only if , in which case the matrix is already diagonal.
The eigenvalues are the same for every : looking at them alone would have given «always diagonalisable», which is false. It is the second mistake in the list above.
Timed Linear algebra practice test.
A test that mixes the topics of the syllabus the way a real exam sitting does, with marking and weak points. It is free and you do not need an account to try it.
Frequently asked questions about Linear algebra.
How do you learn linear algebra when it all seems abstract?
By starting from examples in dimension two and three that can be drawn, and only afterwards from the general definition. A subspace seen as a plane in space stays in your head; the definition with axioms, on its own, does not.
Is Gaussian elimination or the determinant better?
It depends on the exercise, and knowing how to choose is part of the method. In short: Gauss for systems, rank and kernel; the determinant for small matrices and to find the parameter values that make it vanish.
Do you also cover Geometry, with lines, planes and conics?
Yes: in many courses linear algebra and geometry are a single exam, and the analytic geometry of space, with conics and quadrics, is part of the lessons.
How much do proofs matter?
It depends on the lecturer: at the oral a few short proofs are almost always asked. We prepare them like an exercise, understanding which hypothesis each step uses, so they can be rebuilt even if the exact form is forgotten.
The other exams we cover.
- Calculus 1
Limits, derivatives, integrals and series: exam-style exercises and an oral on the hypotheses of the theorems.
See the page → - Calculus 2
Several variables, multiple integrals, fields and surfaces: the sketch comes before the calculation.
See the page → - Physics 1
Mechanics and thermodynamics: the method for setting up the problem, symbols before numbers.
See the page →
To go deeper: rank, kernel and image and eigenvalues, eigenvectors and diagonalisation. For the geometry part there is also the Geometry practice test.
For the full picture go to university tutoring and the maths syllabus.
Make the vocabulary clear, then the calculations.
Message me on WhatsApp with your course and the last exercise that did not work out: we start from there.