Linear algebra tutoring

If linear algebra feels like a different kind of math

It is. Linear algebra is the first course where many students meet mathematics as a study of structures rather than a set of procedures. The mechanics are gentle — you row reduce, you multiply matrices, you take a determinant — but the questions quietly stop being about the mechanics. They become about what a set of vectors spans, whether a system has a solution and why, and what a matrix does when you view it as a transformation.

Students who are drowning in this course are almost never bad at the arithmetic. They are being asked a conceptual question and answering it with a computation, which is why the marks do not match the effort.

What a first linear algebra course covers

Courses begin with systems of linear equations and Gaussian elimination, then reframe those systems as matrix equations. Matrix operations follow — multiplication, the inverse, and elementary matrices — along with determinants and their geometric meaning as a scaling factor for area or volume.

The middle of the course introduces vector spaces and subspaces, linear independence, basis and dimension, and the fundamental subspaces attached to a matrix: column space, null space and row space, tied together by the rank-nullity relationship. Linear transformations then unify the whole picture, with matrices as concrete representations of maps between spaces, and change of basis as the statement that the same map can be written many ways.

Most courses finish with orthogonality — dot products, orthogonal projection, the Gram-Schmidt process, and least-squares fitting — and with eigenvalues, eigenvectors and diagonalisation, including applications such as repeated iteration of a process or decoupling systems. Some courses also cover complex vectors, symmetric matrices, or an introduction to singular values.

Where students commonly get stuck

Span and independence as vocabulary rather than pictures. Students can recite the definitions and still cannot say whether three vectors in three dimensions span the space. We build the geometric picture first — lines, planes, and what "one vector adds nothing new" looks like — and the definitions then stop feeling arbitrary.

Reading a reduced matrix. A row-reduced matrix answers several different questions at once: whether a solution exists, whether it is unique, what a basis for the null space is, and what the rank is. Students who learn only "solve for x" cannot extract the rest, and much of the course is graded on the rest.

Basis versus spanning set versus independent set. The three concepts get blurred, and questions are often designed specifically to see whether they have been distinguished.

Abstract vector spaces. The week polynomials, functions or matrices are declared to be vectors is where a lot of students disengage. It helps enormously to see the axioms as a checklist you can literally verify, one line at a time.

Change of basis. This is the most common single source of confusion in a first course: which direction a change-of-basis matrix goes, and what coordinates mean once the basis is not the standard one.

Eigenvectors computed but not understood. Students find the characteristic polynomial, get roots, solve for vectors, and never register that they have found directions the matrix merely stretches. Diagonalisation then looks like a ritual instead of a simplification.

How sessions work

Sessions are one-to-one, online over Zoom, on an interactive whiteboard. Linear algebra benefits from drawing more than students expect — planes intersecting, vectors being projected, a transformation squashing a square — and we use that heavily alongside your actual assignments and past exams.

We generally alternate between a conceptual question and a computational one, so the definitions stay attached to something concrete. For proof-based courses we spend time on writing arguments straight from definitions, which is usually the difference between a partial mark and a full one.

Which institutions we support

We support students taking linear algebra at UBC and SFU, and we work with students at any institution worldwide.