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CEGEP • Montreal

Linear Algebra and Vector Geometry, 201-SN4-RE (formerly 201-NYC)

Linear Algebra and Vector Geometry, 201-SN4-RE in the revised program (formerly 201-NYC, and Maths 105 in the older coding), is the most abstract of the three mathematics courses of Sciences de la nature. The difficulty is not the arithmetic: it lies in handling objects, vector spaces, that no picture can fully show.

12 fully corrected exercise sets (199 exercises), 12 revision sheets, and a practice paper. Free, no account needed. The same course in French.

Vectors and geometry in space

  1. 1 Vectors in space: the three products Two sets of direct calculations, ten norms, dot products and angles and eight cross and scalar triple products; then components and norm in space, dot product and projection, cross product and areas, scalar triple product and volumes, then an exam-level part: Lagrange's identity, the coplanarity criterion and a complete tetrahedron solved by two methods. Also direction angles and direction cosines with the identity cos²α+cos²β+cos²γ=1 and its trap (no vector is at 45° from all three axes), the orthogonal decomposition of a vector and the distance from a point to a line computed by two matching methods, the vector triple product (BAC minus CAB rule) and the non-associativity of the cross product, and two skew lines classified then measured with the scalar triple product. A Part C adds five classics: the cube in coordinates, a parallelogram in space, decomposing a vector with scalar triple products, the tetrahedral angle of methane and the power received by a tilted solar panel. Sheet
  2. 2 Lines and planes in space One warm-up exercise of ten direct calculations of equations, intersections, distances and angles; then the vector, parametric and symmetric equations of a line, the scalar equation of a plane from its normal vector, the intersection of two planes and the angle between them, relative positions and intersections, point-to-plane and point-to-line distances, then an exam-level part: two skew lines and their distance by the scalar triple product, the paths of two drones that separate crossing paths from a real collision, orthogonal projections and reflections of a point in a plane and in a line, the plane through a line and a point with the pencil of planes, and the bisector planes of two planes. A Part C adds five classics: the angle between a line and a plane, three planes and their three configurations, the common perpendicular of two skew lines, the reflection of a light ray in a plane mirror and the line of steepest slope on a roof. Sheet

Matrices and linear systems

  1. 3 Matrices, systems and determinants Two calculation exercises, ten matrix products and eight inverses to compute directly; then operations and the matrix product, writing a system in matrix form, solving by Gauss-Jordan elimination, the inverse of a 3x3 matrix by the [A|I] method, the inverse matrix and determinants; then an exam-level part: the proof for the transpose of a product, the singular case (zero determinant), a ticket-sales problem solved by Cramer's rule, homogeneous systems and their solution space, the case analysis of a system with parameters by rank, and the properties of the inverse of a product. A Part C adds five classics: matrix equations and the side of the inverse, triangular matrices, the trace of a product, traffic flows in a neighbourhood and a bakery's production in matrices. Sheet
  2. 4 Linear systems, Gauss and Gauss-Jordan One calculation exercise first: eight systems to solve directly by Gaussian elimination, from back substitution to the inconsistent system; then the augmented matrix and elementary row operations, row echelon form by Gauss and reduced row echelon form by Gauss-Jordan, the three cases (unique solution, none, infinitely many) with a parametric solution, inverting a matrix by Gauss-Jordan, then an exam-level part: homogeneous systems and the full discussion of a system with two parameters. Also covered: rank and the Rouché-Capelli consistency theorem, which settles the number of solutions by the rank alone and not by the number of equations, the affine structure of the solution set (general solution equals particular solution plus homogeneous solutions), fitting a parabola through three points with the Vandermonde determinant guaranteeing uniqueness, and balancing a chemical equation as a homogeneous system whose free parameter sets the scale of the reaction. A Part C adds five classics: a zero pivot and a row interchange, a system of four equations, the LU decomposition, the equilibrium temperature of a plate and an ill-conditioned system. Sheet
  3. 5 Determinants and Cramer's rule A calculation drill first: ten determinants to compute directly, from the 2x2 to Cramer's rule. Then 2x2 and 3x3 determinants by Sarrus and by cofactors, an order 4 determinant that exploits its zeros, clever computation by triangularization and common factors, the effect of elementary row operations, the adjugate matrix and the inverse by cofactors. An exam-level part follows: proofs about the determinant, a singular matrix with a parameter, Cramer's rule, the Vandermonde determinant, the determinant as area and volume, and the identity of the adjugate. Part C adds five classics: what multiplicativity rules out, a tridiagonal determinant of order n, lines and planes by determinants, the area of a lot by the shoelace formula and the circle through three points. Sheet

Vector spaces

  1. 6 Linear combinations, linear dependence and bases A computation drill first: ten direct computations of coefficients, coordinates and determinants. Then components and norm in the plane, a linear combination and the system behind it, linear dependence and independence by the determinant, bases and coordinates, and an exam-level part: proofs of independence and uniqueness of coordinates relative to a basis. Also: too many vectors and the extraction of a basis from a spanning set, a complete change-of-basis problem, the properties of subsets and supersets, and true-or-false questions on dimension. Part C adds five classics: resolving a force along two cables, coordinates in an orthonormal basis, a set with a parameter that changes rank, the colours a screen can display and the bases of a crystal lattice. Sheet
  2. 7 Vector spaces and subspaces The step from computation to abstraction: the vector space axioms applied to sets that are not arrows (polynomials, matrices, functions), the subspace test in three checks with the zero vector check that eliminates half the candidates, independence and coordinates in a basis of polynomials, the span and its dimension, the kernel and image of a transformation, and the rank-nullity theorem that ties the three together. Also: the sum and intersection of subspaces with Grassmann's formula, the rank of a matrix and its null space, and differentiation on polynomials. A Part C adds five classics: the direct sum of symmetric and skew-symmetric matrices, the polynomials that vanish at 1, extending an independent set to a basis, two descriptions of the same subspace of R⁴ and the space of linear recurrence sequences. Sheet

Linear transformations and eigenvalues

  1. 8 Linear transformations of the plane A warm-up of ten images to compute directly, from a dilation to a preimage found with the inverse matrix; then the matrices of rotations, reflections, dilations, projections and shears, the image of a point and of a figure, non-commutative composition, the determinant as an area factor, and an exam-level part: proofs of linearity, orthogonal matrices, idempotent projections, a composite transformation solved from start to finish, the reflection about any line, the matrix recovered from non-standard vectors, and the geometric properties that are preserved (lines, midpoints, distances). A Part C adds five classics: three transformations of space, the ellipse that is the image of a circle, undoing a composite transformation, a cast shadow as an oblique projection, and recognizing a transformation matrix. Sheet
  2. 9 Eigenvalues and diagonalization A warm-up of ten eigenvalues to compute directly, from the diagonal matrix to the complex case; then the crowning chapter of the linear algebra course: the relation Av = λv and its geometric reading, the characteristic polynomial and the eigenspaces, the complex eigenvalues of a rotation, the exact diagonalizability criterion with the counterexample that fails it, powers of A computed by diagonalization, symmetric matrices, always diagonalizable, and their orthogonal diagonalization, the Cayley-Hamilton theorem for the inverse and the powers, the eigenvalues of derived matrices, and an application to coupled sequences and Markov chains. A Part C adds five classics: a 3×3 matrix with a double eigenvalue, eigenvalues read off the geometry, diagonalizability depending on a parameter, the normal modes of two carts on springs and the principal axes of an ellipse. Sheet
  3. 10 Matrix models The transition matrix of a Markov chain and its column convention, matrix powers to move several steps ahead, the steady state as a homogeneous system, the Leontief model and its balance equation, the economic reading of the inverse matrix. Then an exam-level part: the Leslie matrix and the population wave, the eigenvalues that govern the limit and the speed of convergence, five statements to correct, mobility between three CEGEP programs, and a published input-output table with a wrong announcement. Part C adds five classics: running a Markov chain backwards in time, the stable structure of a Leslie matrix, the Leontief price model, the students' path as an absorbing chain, and a population that migrates and grows. Sheet

Complex numbers and applications

  1. 11 Complex numbers and De Moivre's theorem Two sets of direct calculations, ten in algebraic form and nine in exponential form with De Moivre's theorem; then algebraic form, conjugate, modulus and quotient, square roots by the algebraic method, trigonometric and exponential forms, De Moivre's theorem and powers, nth roots, followed by an exam-level part: polynomial equations with real coefficients, proofs, the six sixth roots of 64 read as a regular hexagon, complex multiplication as a rotation with the equilateral triangle criterion, the triple-angle formulas and loci in the complex plane. A Part C adds five classics: the fifth roots of unity and cos 72°, the factorization of z⁴ + 4, a quadratic equation with complex coefficients, the iterations of the Mandelbrot set and adding signals with phasors. Sheet
  2. 12 Applications in science Turning a science problem into a linear system, then solving it with the tools of the course: equilibrium of a point under three forces and the determinant of the tension matrix, torque as a cross product and work as a dot product, balancing a chemical equation seen as finding the null space of a matrix, including the redox case where conservation of charge is the missing row, mixing three solutions to a required concentration, Kirchhoff's laws on two loops with the rank read off, shoulder biomechanics, and the centre of mass of weighted points. The thread of the set is the rank: as many pivots as unknowns and the answer is unique, fewer pivots and a constraint is missing, an impossible row and it is the situation described that is impossible. Three exercises end on an inconsistent system whose conclusion is physical rather than mathematical, and a fourth on a consistent system that nevertheless has no admissible solution. Sheet

When the chapters are done

A full paper under exam conditions, on the whole course. Do it once every chapter is behind you: it measures what you can do on a question you have never seen.

Practice exam

Practice final exam, 201-SN4-RE

A three-hour final exam that runs through the whole course in linear algebra and vector geometry: the three products, lines and planes, systems and matrices, determinants, rank, vector spaces, transformations of the plane, diagonalization and complex numbers. No question tells you which method to use.

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

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