How to Study Linear Algebra: Definitions Do the Work

Study linear algebra as a definitions course. Definitions carry the work, so learn to state span, linear independence, basis, rank and null space from memory and then use each one on a real matrix. Treat clean row reduction and knowing what the answer means as two separate skills, because a course examines both.

School algebra rewards fluency with the rules and calculus rewards volume of problems worked by hand, which our guides to how to study algebra and how to study calculus cover in full. Linear algebra keeps that computation and adds a layer above it, where the question is whether you can apply a stated property to an object you have never seen.

What does a linear algebra course actually contain?

Courses differ, so take one published syllabus as a map and check it against your own. MIT OpenCourseWare publishes 18.06 Linear Algebra, taught by Prof. Gilbert Strang, in a Spring 2010 version. Its course description names the territory in two sentences: “This is a basic subject on matrix theory and linear algebra. Emphasis is given to topics that will be useful in other disciplines, including systems of equations, vector spaces, determinants, eigenvalues, similarity, and positive definite matrices.”

The syllabus lists one prerequisite, “Multivariable Calculus (18.02)”. That is how MIT sequences its own degree, and many universities place linear algebra earlier, so check where yours sits. The set text is Strang, Introduction to Linear Algebra, Wellesley-Cambridge Press, in the 4th edition of 2009 and the 5th of 2016.

The goals section states the ambition of the course in one line: “The goals for 18.06 are using matrices and also understanding them.” Every entry underneath pairs a computation with the idea it carries. Three of them show the arc:

• “Complete solution to Ax = b (column space containing b, rank of A, nullspace of A and special solutions to Ax = 0 from row reduced R)”
• “Basis and dimension (bases for the four fundamental subspaces)”
• “Eigenvalues and eigenvectors (diagonalizing A, computing powers A^k and matrix exponentials to solve difference and differential equations)”

The list runs on into least squares and projections, Gram-Schmidt and the factorization A = QR, determinants, symmetric matrices and the singular value decomposition. The four fundamental subspaces in the second bullet are the standard organising picture for the middle of the course: the column space and null space of A, plus the row space and left null space, which are the column space and null space of A transpose. Questions about solving Ax = b keep asking where b sits in relation to those four.

What makes linear algebra different from earlier maths?

The arithmetic gets easier: elimination is addition and multiplication. What changes is that the objects are defined by properties, so a question stops asking how to compute something and starts asking whether a stated property holds.

Take the commonest question of the first half: are these vectors linearly independent? The only route in is the definition, that the sole linear combination of them equal to the zero vector is the one with every coefficient zero, which you turn into a homogeneous system and solve. The mark is for the argument the definition licenses, and the row reduction is one step inside it.

So keep a definitions page and treat it as examinable in its own right: span, linear independence, basis, dimension, rank, null space, column space, eigenvalue and eigenvector. Write each out from memory, then apply it that same sitting to a matrix in front of you. Definitions behave much like formulas here, being small, exact and quickly lost without retrieval, so the habits in our guide to how to memorize formulas carry across.

Proof arrives as a routine expectation

For many students this is the first course where a short proof is a weekly requirement. The reassuring part is that introductory proofs here are mostly definition unfolding. Write out the definition of every term in the statement first, and a proof that looked like a blank page becomes a checklist.

Three moves cover much of the rest: take an arbitrary element and show it has the property, assume a linear combination equals the zero vector and show each coefficient must vanish, or count dimensions. School geometry is where most students first met the discipline of naming a reason for every line, and our guide to how to study geometry sets that habit out; a linear algebra proof is graded on the same discipline.

Why does the lecture feel clear and the problem set not?

The lecture made sense, the derivation followed, and then the first question on the problem set refused to start. There is measured evidence that the feeling of learning is an unreliable guide to the learning itself.

Measuring actual learning versus feeling of learning in response to being actively engaged in the classroom, by Louis Deslauriers, Logan S. McCarty, Kelly Miller, Kristina Callaghan and Greg Kestin, appeared in PNAS in September 2019 (volume 116, issue 39, pages 19251 to 19257). Its abstract reports that “Students in active classrooms learned more (as would be expected based on prior research), but their perception of learning, while positive, was lower than that of their peers in passive environments.” The authors put it more bluntly in the text, writing that “students in the active classroom learn more, but they feel like they learn less”, and they name a mechanism: “We show that this negative correlation is caused in part by the increased cognitive effort required during active learning”.

Read the scope carefully. That study ran in large introductory college physics and compared active instruction with passive lectures. It tested no linear algebra, and no technique a student performs alone at a desk. Carrying it across to your own course is this guide reasoning by analogy, on one narrow point: the smoothness of a lecture you are following is weak evidence about what you can produce without it. The cheap test is to close the notes after a class and rebuild one result from the definitions up.

Computation and concept are graded apart

The other half of the gap is that a lecture demonstrates the computation while the problem set asks for the reading. Both are examined, and each answers to its own kind of practice.

Why does the same fact keep coming back with a new name?

A square matrix is invertible or it is not, and that single fact gets stated in at least five ways, each arriving in a different week attached to a different topic.

For a square matrix those five lines are one fact in five costumes, and any one hands you the other four. Build them as a single card and rehearse in both directions: given one face, name the rest; given a zero determinant, say what that forces about the columns, the rank and the solutions of Ax = b. The rank plus nullity relation earns the same treatment: for a matrix with n columns, the rank plus the dimension of the null space equals n.

How should you practise linear algebra?

Three routines carry most of the load here.

State the definition, then spend it. Take a blank page, write one definition from memory, turn the page over, and answer a question that needs it. Recall alone leaves you reciting a definition you cannot use, and application alone lets you lean on a page you will not have in the exam. Pairing them is what makes a definition operational. Space the rehearsals across the term on a spaced repetition schedule, because the vocabulary of week two is what week nine assumes you own.

Run one matrix through several questions. Take a single 3 by 4 matrix and ask everything of it in one sitting: row reduce it, name its rank, give a basis for its column space, give a basis for its null space, notice that its four columns cannot be independent, since four vectors in three dimensional space never are, and state what that forces about the solutions of Ax = b. One matrix, six answers, each checking the others.

Sort every lost mark into one of three bins. A wrong answer here has three very different causes, and each has its own repair.

• Arithmetic. A sign lost in an elimination step, with the method visibly right around it. Slow the bookkeeping down and verify each pivot. Theory revision does nothing for this bin.
• Definition recall. You could not state what the question was asking about. Back to the definitions page and to retrieval practice.
• Method choice. You knew every definition involved and could not see which one opened the question. Mixed problem sets with the chapter headings stripped off are the practice that moves this bin.

Keep the tally for a fortnight. If the bins come out uneven, check whether your studying is aimed at the smallest one.

Where linear algebra marks are actually lost

Where GeniusPal fits in a linear algebra routine

The three bin tally shows where a question generator helps. Arithmetic and method choice are trained by working problems on paper. Definition recall is the bin GeniusPal can drill. Point it at a file you already have, up to 10 MB as a PDF, Word, PowerPoint, text, Markdown or CSV document, or at a link to a page, and it writes questions with answers from that material.

GeniusPal does no mathematics: it will not solve a system, row reduce a matrix, compute an eigenvalue, or mark a page of working, so every computation above stays yours. There is no optical character recognition either, so a scanned problem sheet with no text layer will not read.

What it fits is the definitions page and the five faces above: which property defines a basis, what the rank counts, and what the null space holds. The daily review costs nothing on any plan and gathers up to 20 questions from across every set you own, putting your misses first, then anything you have never answered, then whatever is short of mastery.

Free gives 2 generations for the lifetime of the account, sets of 10 questions, and 2 complete runs of each set, with quizzes that are multiple choice throughout. A paid plan adds flashcards, written active recall, typed short answers mixed into a quiz, and sets of up to 30 questions. Student is $14.99 a month with 100 generations. Genius is $59.99 a year, displayed as $5.00 a month, under a fair use ceiling.