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MATH 133 Tutor in Montreal, McGill Linear Algebra and Geometry

MATH 133 is examined by hand, and that is the first thing to get right about it. The numbers on a midterm are chosen to row reduce cleanly, the marks go to each row operation you name, and the vector spaces never leave Rn. Most linear algebra you find by searching was written either for a calculator or for the abstract second course, so it skips the steps your marker is looking for or wanders into material this course never examines. Everything on this page is written for MATH 133 as it is actually examined.

14 chapters, 171 corrected exercises and 14 revision sheets, plus 2 practice papers. Free, no account needed: the full worked solution is on the page.

Linear systems and matrices

  1. 1 Systems of linear equations and Gaussian elimination Twelve systems to solve by Gaussian elimination to warm up, then ten exercises on the first chapter of MATH 133, built on one rule: a row operation is legal only if it can be undone, and the answer is read off the echelon form and nowhere else. Part A is the mechanics: copying a system into an augmented matrix, the legal and illegal moves, two echelon forms that reduce to the same reduced form, free variables and the general solution, and two systems with a parameter k whose trap is a division by an expression that can vanish. Part B works at exam level: three lines in the plane, polynomial interpolation, five statements to correct, a circuit solved with Kirchhoff's laws, a balanced chemical equation and a fertilizer mix. No calculator anywhere, as on the exam. Sheet
  2. 2 Homogeneous systems and the rank theorem Ten exercises on the chapter where a system stops being solved and starts being counted. Part A: deciding whether a homogeneous system has a nontrivial solution before reducing anything, basic solutions and why there are exactly as many as free columns, the rank theorem applied to wide, tall and square matrices, the product Ax read as a combination of the columns, and consistency decided by a condition on the right-hand side. Part B: every solution as one particular solution plus the homogeneous part, linearity used on a matrix you are never shown, five statements to correct, balancing chemical equations, and an order filled from four feed mixes. No calculator needed anywhere. Sheet
  3. 3 Matrix algebra and the matrix product Ten exercises on the chapter where a matrix stops being a table of numbers and becomes something you multiply. Part A builds the product: the sizes checked before any entry, one entry or one column computed without the rest, a product that is zero in one order and not in the other, the transpose and its reversal rule, symmetric and skew-symmetric parts, and the powers of nilpotent and idempotent matrices. Part B works at midterm level: every matrix that commutes with a given one, the trace and why AB minus BA is never the identity, five statements to correct, a price and quantity table, and flight routes counted with the square of an adjacency matrix. No calculator anywhere, as the course requires. Sheet
  4. 4 Matrix inverses and elementary matrices Ten exercises on the chapter where a matrix gets undone, and where the side you undo it from is the whole answer. Part A builds the tools: the definition checked on both sides and the 2 by 2 shortcut, the [A | I] algorithm run cleanly, run into a matrix with no inverse, and run with a parameter, then the rules for the inverse of a product, a transpose, a multiple and a power, and the sum that has no rule. Part B works at exam level: elementary matrices and the order of their product, matrix equations isolated on the correct side, five statements to correct, a matrix code decoded from the left and from the right, and the proofs where a polynomial identity hands you the inverse. Sheet

Determinants

  1. 5 Determinants: cofactor expansion and row operations Ten exercises on computing a determinant without brute force. Part A builds the two tools: the checkerboard of signs and the minor that is not the cofactor, a 4 by 4 with a nearly empty column, reduction to a triangle with the swaps counted, the ledger of factors pulled out of rows and columns, and a known determinant from which every other one follows. Part B works at exam level: a determinant with a parameter factored before it is expanded, a 4 by 4 solved by mixing both tools, five statements to correct, the Vandermonde determinant behind a parabola through three measurements, and the band matrix of a chain of springs. No calculator anywhere, as the course requires. Sheet
  2. 6 Determinants, inverses and Cramer's rule Ten exercises on the chapter where the determinant stops being a computation and becomes a tool. Part A: a composite determinant evaluated from four rules without a single entry, the power n that turns the determinant of 2A into eight times that of A, the values of a parameter that switch invertibility off, the adjugate built from nine cofactors, and one entry of the inverse and one unknown of a system read off by Cramer's rule. Part B works at exam level: short proofs by taking determinants, the adjugate identity used as a tool, five statements to correct, a voxel remap undone exactly by an integer inverse, and a circuit where only one current is needed. No calculator, as the course requires. Sheet

Vector geometry in space

  1. 7 Geometric vectors and lines in space Ten exercises on the chapter where linear algebra becomes geometry. Part A builds the tools: a vector as a displacement and the parallelogram rule, norms, parallel vectors and the point two thirds of the way along a segment, the three ways to write one line and the conversions that go wrong, then the relative position of two lines in space, posed three times with its three traps. Part B works at exam level: vector proofs on parallelograms and midpoints, the centroid of a triangle, five statements to correct, two drones whose paths cross without a collision, and a final exam problem with a parameter. Sheet
  2. 8 Dot product, projections and planes Ten exercises on the chapter where every length and every angle turns out to be a projection. Part A builds the tools: the dot product and the angle it hides, the projection split into a shadow and a leftover, the distance from a point to a line with its closest point, a plane written from one point and a normal, and the three ways a line can meet a plane. Part B works at exam level: the distance from a point to a plane derived rather than recited, parallel planes and the acute angle between two planes, five statements to correct, work as a dot product on a cart hauled up a track, and a roof whose pitch, lamp, shadow and solar panel are all read off one normal vector. Sheet
  3. 9 Cross product, areas and volumes Ten exercises on the one product of the course that answers with a vector, whose direction is a normal and whose length is an area. Part A builds the tools: the component formula and the ten-second check on the middle sign, the length through the sine and Lagrange's identity, triangles built from edges rather than position vectors, the plane through three points posed three ways, and the triple product with its one sixth and its coplanarity test. Part B works at exam level: the line where two planes meet, the distance from a point to a line read as a height, five statements to correct, the torque on a wrench, and the distance between two skew lines with their common perpendicular. Every number reduces by hand. Sheet

Subspaces, transformations and eigenvalues

  1. 10 Subspaces, span and linear independence Ten exercises on the chapter where linear algebra starts asking for proofs. Part A drills the three-point subspace test on planes, shifted lines, quadrants, pairs of axes, squares, absolute values and a parameter, proves that the null space and the image are subspaces, turns every span question into one system, and makes independence hand over its dependency relation. Part B reads a span as a line, a plane or all of space, hunts the values of k that break independence, corrects five statements, mixes protein powders that the algebra allows and the kitchen does not, and ends on the classic final-exam proof about u + v, v + w and u + w. Sheet
  2. 11 Basis, dimension and the fundamental subspaces Ten exercises on the chapter where row reduction stops being a way to solve systems and becomes a way to count. Part A builds the tools: a plane with two bases and one dimension, a basis extracted from a spanning family, the null space, column space and row space of one matrix read from one reduction, the same reading with a row swap hidden inside, and rank plus nullity used to conclude from the size alone. Part B works at final-exam level: completing an independent family into a basis, subspaces cut out by equations, five statements to correct, coordinates in a crystal lattice, and a matrix with a parameter whose rank drops. Sheet
  3. 12 Linear transformations and their matrices Ten exercises built on one gesture: a linear transformation is decided by where it sends the standard basis, and those images are the columns of its matrix. Part A proves and disproves linearity with numbers, builds the standard matrix from a formula, from a rotation, a shear and a stretch, from a reflection and a projection across a line, and from images of vectors that are not the standard basis. Part B works at final exam level: composition and the order of the factors, inverses and the area factor read on the unit square, five statements to correct, a 3D graphics pipeline with projections and reflections through a plane, and a colour sensor whose kernel explains why two different lights can look identical. Sheet
  4. 13 Eigenvalues and eigenvectors Ten exercises built on one idea: every eigen question is a question about the matrix x I minus A, whose determinant finds the numbers and whose null space finds the directions. Part A is the mechanics: testing a given vector with one multiplication, the 2 by 2 polynomial and its eigenspaces, a 3 by 3 expanded along its row of zeros, reflection, projection and rotation read off the picture, and a 4 by 4 block triangular matrix whose lower eigenvectors refuse to pad with zeros. Part B works at final exam level: eigenvalues of powers, inverses, shifts and transposes, the trace and the determinant used to find a missing eigenvalue, five statements to correct, a stretched rubber sample, and a companion matrix whose eigenvectors are (1, t, t squared). No calculator anywhere, as the course requires. Sheet
  5. 14 Diagonalization and its applications Ten exercises on the last chapter of the course, where eigenvectors stop being a computation and become a change of axes. Part A settles the classic question: two matrices one sign apart, one diagonalizable and one not, a triangular matrix whose verdict depends on one parameter and not the other, P and D built in matching order, powers of a matrix read in the eigenvector axes, and similar matrices with the counterexample that shares every invariant. Part B works at final exam level: a bird population with two age classes, a bike-sharing Markov chain and its steady state, five statements to correct, a paving count solved as a recurrence, and a predator and prey model whose dominant eigenvalue crosses 1. Every polynomial factors by hand, as the course requires. Sheet

The whole course in one set

Ten exercises that cross every chapter, built on one idea: nearly every question in MATH 133 is a question about the rank. Best done after the chapters, as the review that ties them together.

Linear systems, matrices and rank, MATH 133 at McGill

Ten exercises built on one idea: every question in a first linear algebra course is a question about the rank. Part A is the mechanics, row reduction and the staircase of pivots, why a system never has exactly two solutions, matrix algebra and the identities that fail, the determinant read as a collapsing area, independence and spanning as one pivot count. Part B works at midterm level: the invertible matrix theorem used as a tool rather than recited, three planes in space and the four ways they meet, five statements to correct, a traffic network whose dependency is forced by conservation, and eigenvalues arriving as one more singularity test.

10 corrected exercises →

When the chapters are done

Full papers under exam conditions, without a calculator. None of their questions repeats one already solved in the chapter sets above: the situations are new, so they measure what you can do rather than what you remember reading.

Practice exam

Practice midterm, MATH 133 at McGill

A ninety minute practice midterm for MATH 133, eight questions and one hundred points, on the first six chapters of the course, with the full solution under every question. Part A asks four short questions: a circle through three measured points found by elimination, basic solutions and the right-hand sides that make a system consistent, two rectangular matrices and the polynomial their product satisfies, and five statements to judge true or false. Part B asks four long problems: a system whose parameter sits in two places, a 3 by 3 inverse used on the correct side, a 4 by 4 determinant with its consequences and an area on a grid, and the temperatures inside a heated plate found by Cramer's rule and the adjugate. No calculator.

  • 8 questions
  • 100 points
  • 90 minutes
Sit the paper

Practice exam

Practice final exam, MATH 133 at McGill

A three hour practice final for MATH 133, twelve questions and one hundred points on the whole course, weighted like a real cumulative final, with the full solution under every question. Part A weighs systems, matrices and determinants: a system with two parameters, an inverse used through its transpose, and Cramer's rule with one entry of the inverse. Part B weighs the geometry of space: two lines that meet and a parallel that does not, a point and a plane with distance, foot and mirror image, a plane through three points and a tetrahedron. Part C weighs subspaces and linear maps: two sets to test and a family with a parameter, the four fundamental subspaces of one matrix, and a map known on three vectors with its kernel and a right inverse. Part D weighs eigenvalues: a 3 by 3 whose polynomial arrives factored, a repeated eigenvalue with a power computed without P inverse, and two lakes joined by a channel under a fishing quota. No calculator.

  • 12 questions
  • 100 points
  • 180 minutes
Sit the paper

What MATH 133 covers

The official description reads like a list: systems of linear equations, matrices, inverses and determinants; geometric vectors in three dimensions, the dot product, the cross product, lines and planes; an introduction to vector spaces, linear dependence and independence, bases; linear transformations; eigenvalues and diagonalization. It is a three-credit course, and it is closed to anyone holding the CEGEP linear algebra objective, 00UQ. That restriction tells you who is in the room: students meeting matrices for the first time, many of them in their first term at McGill.

What the list hides is that it is one course, not 14 separate topics. A single row reduction decides how many solutions a system has, whether a matrix is invertible, whether vectors are independent, what the dimension of a subspace is and whether a number is an eigenvalue. Students who learn the chapters as separate recipes run out of time on the final; students who see the same computation behind each question do not.

The chapters above follow the order of the course and of its textbook, Nicholson’s Linear Algebra with Applications. Each one has a corrected exercise set, with the full solution written out on the page, and a revision sheet that answers a different question: not what the chapter says, but what loses marks on it.

Where MATH 133 marks are actually lost

Row reducing without writing the operations

A chain of matrices with no operations between them cannot be checked, so one slip in the second step costs every mark after it. Write each operation, one per arrow, and a marker can follow you past the mistake and give you the method.

Treating matrices like numbers

Expanding a square as though the product commuted, cancelling a matrix that is not invertible, writing the inverse of a sum as a sum of inverses. Every rule you import from ordinary algebra has to be checked against the one fact that changes everything: the order of a product matters.

Reading the answer off the wrong matrix

The pivots are found on the reduced matrix, but a basis of the column space is taken from the original columns. Determinants change under row swaps and scalings, and forgetting the factor is the most common way to lose a determinant question that was otherwise done right.

Saying “no” without a counterexample

To show a set is not a subspace, or a map is not linear, one explicit failing example is the whole proof, and without it the answer is a guess. To show that it is, the argument must hold for every vector, not for the two you tried.

Frequently asked questions

What is MATH 133 at McGill?

MATH 133, Linear Algebra and Geometry, is McGill University’s three-credit first course in linear algebra. It covers systems of linear equations, matrices, inverses and determinants; geometric vectors in three dimensions, the dot and cross products, lines and planes; an introduction to vector spaces through the subspaces of R^n, linear independence and bases; linear transformations; and eigenvalues and diagonalization. It is a required first-year course across science, engineering and several management programs.

I took linear algebra in CEGEP. Do I need MATH 133?

Usually not. MATH 133 is closed to students who have CEGEP objective 00UQ or its equivalent, which is the CEGEP linear algebra and vector geometry course. The students in the room are therefore mostly first-year students from outside Quebec, international students, U0 students, and students whose CEGEP program did not include linear algebra: most of them are meeting matrices for the first time.

Can I use a calculator on MATH 133 exams?

Assume you cannot: MATH 133 midterms and finals have typically been written without calculators, so check the rule on your own course outline. That is why every exercise on this page is built with numbers that reduce cleanly by hand, and why every solution names each row operation. A method the marker can follow earns the marks even when an arithmetic slip spoils the final number.

What is the difference between MATH 133 and MATH 223?

MATH 133 works entirely in R^n: its vector spaces are lines, planes and their higher-dimensional cousins, and everything is computed with matrices. MATH 223, the next course in linear algebra, moves to abstract vector spaces, inner products, orthogonal diagonalization and the spectral theorem. The chapter sets here stay inside MATH 133 on purpose: no polynomial spaces, no Gram-Schmidt, no complex eigenvalues.

Do you offer MATH 133 tutoring in Montreal?

Yes. I tutor MATH 133 and the other first-year McGill math courses in Montreal, in person and online. I am a McGill graduate and I work on your own course outline, assignments and past midterms rather than on a generic syllabus.

Other first-year courses

The same free corrected material exists for other courses, some of it in English and the rest in French.

COMP 202, Foundations of Programming (McGill)PHYS 101, Introductory Mechanics (McGill)MATH 140, Calculus 1 (McGill)MATH 141, Calculus 2 (McGill)MATH 203, Calculus I (Concordia) Math tutoring in Montreal All revision sheets in English Higher education, in French
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