How to Pass Linear Algebra: A Topic-by-Topic Plan for Computations and Proofs

How to pass linear algebra: drill the computations (row reduction, matrix multiplication, determinants, eigenvalues) until they're fast and accurate, learn every definition exactly as your textbook states it, and practice the conceptual questions on their own, because computing alone won't answer them. Read each matrix as columns and as a transformation so the theorems connect, and quiz yourself on definitions and missed problems on a schedule instead of rereading notes.

This guide covers what a first linear algebra course includes (checked against MIT's and Penn State's course pages), why it feels different from calculus, a topic-by-topic plan with a free place to practice each one, how to approach proofs, and how to study for a linear algebra exam.

What linear algebra covers

The core topics recur from school to school, even when the order changes. MIT's linear algebra course, taught by Gilbert Strang and free on OpenCourseWare, runs in three units: solving Ax = b and the four fundamental subspaces; least squares, determinants and eigenvalues; then positive definite matrices and applications (MIT OpenCourseWare). Its readings come from Strang's textbook, Introduction to Linear Algebra, now in a 6th edition from 2023. Penn State's MATH 220 lists systems of linear equations, matrix algebra, eigenvalues and eigenvectors, and linear systems of differential equations, with calculus as the prerequisite (Penn State).

Put together, a typical first course looks like this:

1. Systems of equations and row reduction: Gaussian elimination, echelon forms, pivots, free variables.
2. Matrix algebra: multiplication, inverses, transposes, elementary matrices, LU factorization in some courses.
3. Vector spaces: span, linear independence, basis, dimension, subspaces, rank and null space.
4. Linear transformations: matrices as maps, kernel and range, change of basis.
5. Determinants: computing them and what they say about invertibility.
6. Eigenvalues and eigenvectors: the characteristic polynomial, diagonalization.
7. Orthogonality and least squares: dot products, projections, Gram–Schmidt, best-fit solutions.

Engineering and data-science sections often lean applied. Boyd and Vandenberghe's Introduction to Applied Linear Algebra, free online, covers vectors, matrices and least squares and is the textbook for Stanford's ENGR108 and UCLA's EE133A (Stanford). Math-major sections lean on proofs. Your syllabus tells you which kind you're in.

Is linear algebra hard?

It's hard in a different way than calculus. Calculus rewards technique on a problem type you recognize. Linear algebra adds a layer of abstraction early: MIT's course reaches the four fundamental subspaces in its first unit, and the question shifts from "row-reduce this matrix" to "is this set a subspace, and why?" Three things trip people up:

• Vocabulary that has to be exact. Span, basis, independence and rank have precise meanings. A fuzzy definition gives a wrong answer to a true-or-false question that looks easy.
• Proofs, sometimes for the first time. Princeton's page for its honors linear algebra course answers students for whom it's a first course with rigorous proofs, and tells them to be ready to spend at least 10 hours a week working outside class (Princeton). Non-honors courses ask for less, but expect at least a few short arguments.
• Arithmetic that compounds. One sign slip in row reduction ruins every later step. Build the habit of checking: multiply A by your answer and compare the result with b before you move on.

The good news: the ideas reuse each other constantly. Once rank, null space and invertibility click, many of the course's theorems turn out to be versions of one statement, the invertible matrix theorem.

How to study linear algebra

1. Make definitions flashcards on day one. Front: the term. Back: the definition, word for word from your textbook, plus one example and one non-example. Review them every few days for the whole term.
2. Draw the picture. For any 2×2 matrix, sketch what it does to the unit square; the free Matrix Transformation Visualizer lets you check your sketch for rotations, shears and projections. For Ax, think of x as the recipe and A's columns as the ingredients.
3. Separate computation drills from concept questions. Do ten row reductions in a row until they're automatic, then switch to "explain why" questions. Mixing both kinds in one session works only once the mechanics are solid.
4. Explain each step out loud. Saying why each row operation is allowed, or why a set is or isn't independent, is self-explanation, which a major review of study techniques rated moderately useful; it rated practice testing and spreading study out as the most useful of the ten it examined, and rereading as among the least (Dunlosky et al., 2013).
5. Redo, don't review. When you miss a problem, cover the solution and redo it from a blank page two days later.

How to pass linear algebra: a topic-by-topic practice plan

Lykke's free Applied Linear Algebra and Least Squares course has quizzes, flashcards and interactive demos for the applied core. Its nine sections don't cover determinants, subspaces or eigenvalues, so the table fills those gaps from the free Precalculus course and from short readings.

Rusty on matrix basics from high school? The same Precalculus section on matrices also reviews multiplication, inverses and solving systems, and the precalculus guide covers the algebra the course assumes. For more problems with full solutions, MIT's OpenCourseWare version of the course posts problem sets and exams with solutions (MIT OpenCourseWare).

Four worked checks you should be able to do in a minute

Each one takes about a minute once the method is automatic. Do each by hand, then check it.

• Solve a system by row reduction. For x + 2y = 5 and 3x + 4y = 6, subtract 3 times row 1 from row 2: −2y = −9, so y = 4.5 and x = 5 − 9 = −4. Check in the second equation: 3(−4) + 4(4.5) = 6.
• Use a determinant. The matrix with rows (1, 2) and (3, 4) has determinant 1·4 − 2·3 = −2. It isn't zero, so the matrix is invertible and the system above has exactly one solution.
• Find eigenvalues. For the matrix with rows (2, 1) and (1, 2), solve (2 − λ)² − 1 = 0 to get λ = 1 and λ = 3. The eigenvector for 3 is (1, 1); for 1 it's (1, −1). Multiply to confirm: the matrix sends (1, 1) to (3, 3).
• Test a subspace. The plane x + y + z = 0 in three dimensions contains the zero vector and stays closed under addition and scaling, so it's a subspace. The plane x + y + z = 1 doesn't contain the zero vector, so it isn't.

How to handle linear algebra proofs

Proofs in a first course are usually short, and most start the same way: write down the definition of what you're asked to show, then write down what you're given, and look for the bridge.

1. Translate the claim into definitions. "Show the columns are independent" means: show the only solution to c₁v₁ + … + cₙvₙ = 0 is all c's equal to zero.
2. Name your objects. Let u and v be vectors in the set; let c be a scalar. Then check each condition with those names.
3. Use the big equivalence. For a square matrix, being invertible, having a nonzero determinant, having independent columns, having only the zero solution to Ax = 0 and having no zero eigenvalue all go together. Many proofs and true-or-false questions are one step along that chain.
4. Find a counterexample for false statements. A single 2×2 matrix usually settles it. Keep a few handy: the zero matrix, the identity, and a matrix with a repeated row.

How to study for a linear algebra exam

• A week out: list every definition and theorem on the exam. Quiz yourself on all of them, without notes, and star the misses. Testing yourself pays off later: in a classic experiment, students who were tested on a passage retained substantially more two days and a week later than students who restudied it (Roediger & Karpicke, 2006).
• Days 6–4: one computation drill per day (ten problems, timed), then redo every missed homework problem from a blank page.
• Days 3–2: work through a past exam under a timer. MIT's open course has exams with solutions if your professor doesn't post old ones. Grade it honestly and fix the topic behind each lost point.
• The day before: definitions, the invertible-matrix chain and your starred misses. Then sleep.

Definitions and theorems are exactly the kind of material that sticks with spaced review; the spaced repetition guide shows how to schedule it. If you took calculus last year, the Calculus 1 guide covers study habits that carry straight over to this course.

If you're behind in linear algebra

Find the earliest topic you can't do without notes, usually row reduction or independence, and fix that first; everything later depends on it. Spend one session on its definitions and five problems, then move forward a topic at a time. Go to office hours with a specific problem you got wrong and your attempt at it, so the professor can see where a definition went wrong. A study group helps too, as long as each person works the problem before you compare; Princeton points its students to office hours, problem sessions and study groups. Explaining a step to a classmate shows quickly whether you understand it. If a midterm already went badly, the guide on what happens if you fail a midterm walks through the grade math and the next steps.