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McGill University · MATH 133

McGill MATH 133: Linear Algebra and Geometry tutoring

Linear Algebra and Geometry: one-to-one online tutoring and exam preparation.

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Course overview

McGill MATH 133 connects systems and matrices with three-dimensional geometry, vector spaces, transformations, eigenvalues, and diagonalization. The course combines calculations with increasingly abstract definitions.

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MATH 133 topics to review

Systems and matricesInverses and determinantsVectors in three dimensionsVector spaces and basesLinear transformationsEigenvalues and diagonalization

Common difficulties

  • Relating pivots and free variables to a solution set
  • Connecting three-dimensional geometry with matrix calculations

Preparing for this course

Use equations, fractions and coordinates fluently. Explain what a simultaneous solution means before using a matrix algorithm.

Registration requirements

The guidance above is study preparation, not an eligibility decision. Confirm current prerequisites, restrictions and availability with McGill University.

Check the official course catalogue

Try a course problem

Find the real unknowns x,yx,y satisfying both x+y=7x+y=7 and 2x−y=22x-y=2.

Show the reasoning

Add the equations to eliminate yy: 3x=93x=9, hence x=3x=3. Substituting into x+y=7x+y=7 gives y=4y=4. Check both originals: 3+4=73+4=7 and 2(3)−4=22(3)-4=2.

When three vectors span only a plane

A system can have three unknown coefficients without giving three independent directions. This example connects linear combinations, consistency and the choice of a basis.

1. Look for a relationship before reducing a matrix

Work in three-dimensional real coordinate space. Each displayed triple lists a vector’s first, second and third coordinates. Let u=(1,0,1)u=(1,0,1), v=(0,1,1)v=(0,1,1) and w=(1,1,2)w=(1,1,2). Notice that w=u+vw=u+v. The relation u+v−w=0u+v-w=0 uses coefficients that are not all zero. This proves that the three vectors are linearly dependent. It does not mean that every linear combination is the zero vector.

2. Describe every vector in the span

Let a,b,ca,b,c be arbitrary real coefficients multiplying u,v,wu,v,w, respectively, and let (x,y,z)(x,y,z) denote a target vector. The span is the set of all combinations obtained by varying those coefficients. An arbitrary combination is au+bv+cw=(a+c, b+c, a+b+2c).\begin{gathered}a u+b v+c w\\=(a+c,\,b+c,\,a+b+2c).\end{gathered} Its third coordinate equals the sum of its first two. Conversely, any vector (x,y,x+y)(x,y,x+y) equals xu+yvx u+y v. Therefore the span is exactly the plane z=x+y.z=x+y.

3. Check whether a representation exists and whether it is unique

The vector (2,3,5)(2,3,5) lies in the plane and equals 2u+3v2u+3v. Choose any real number tt as the coefficient of ww. Since w=u+vw=u+v, subtracting tt from the other two coefficients leaves the target unchanged. This gives infinitely many representations: (2,3,5)=(2−t)u+(3−t)v+tw,t∈R.\begin{gathered}(2,3,5)\\=(2-t)u+(3-t)v+t w,\\t\in\mathbb R.\end{gathered} In contrast, (2,3,6)(2,3,6) has no representation because 6≠2+36\ne2+3. Dependence can give nonunique coefficients for vectors in the span; it does not make every target attainable.

4. Choose a basis and explain the dimension

The pair u,vu,v is independent: the first two coordinates of au+bv=0a u+b v=0 force a=b=0a=b=0. Since this pair also spans the plane, {u,v}\{u,v\} is a basis and the plane has dimension 22. Coordinates in this basis are unique.

Try it yourself

Add p=(0,0,1)p=(0,0,1). Is {u,v,p}\{u,v,p\} a basis of R3\mathbb R^3? Express an arbitrary (x,y,z)(x,y,z) using this set.

Show the explained solution

Yes. The new vector is outside the plane z=x+yz=x+y, and (x,y,z)=xu+yv+(z−x−y)p.\begin{aligned}&(x,y,z)\\&=x u+y v\\&\quad+(z-x-y)p.\end{aligned} The first two coordinates force the coefficients of u,vu,v to be x,yx,y; the third then fixes the coefficient of pp. Thus the representation exists and is unique for every target.

For your next vector problem, decide which question is being asked: does a representation exist, is it unique, or what is a basis? These require related but different explanations.

How tutoring can help

Your tutor can move through your system, augmented matrix and geometric sketch with you to locate the first unclear step. Contrasting an inconsistent system with a consistent system that has a free variable can help you explain the difference instead of relying only on row-reduction rules.

Exam preparation for MATH 133

After each row reduction, write one sentence explaining the result. Mix calculations, geometry and definitions, and check proposed solutions by substitution.

Use your section’s outline to choose the chapters to review. We work through permitted practice and comparable problems, without completing graded assignments or exams on your behalf.

Useful links for preparation or further study—not transfer equivalencies or a required registration sequence.

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Independent practice

Free practice and study guides for MATH 133

8 min

Row Operations: Why the Solutions Stay the Same

Understand reversible row operations, solve and check a linear system, and distinguish one, no and infinitely many solutions. Original practice for Concordia MATH 204 and McGill MATH 133.

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