Concordia MATH 204: Tutoring & Worked Practice

Listed course

Course overview

Concordia MATH 204 connects linear systems and matrix methods with vector geometry. Its Fall 2026 outline also includes vector spaces, subspaces, bases, dimension, eigenvalues and diagonalization. Use your section’s current outline to choose the topics to review.

Your MATH 204 revision route

The outline’s scheduled November 1 midterm date is now past. Keep weeks 1–6 practice for systems, matrices and basic vectors, then add later-course topics for the whole-course final. The selected lessons below do not cover the complete final syllabus; check your current section instructions and official exam schedule.

Choose topics. Separate early-course practice from later-course revision
Follow a worked example. Find basis coordinates, then convert between two bases
Try without solutions. Try six progressive coordinates and basis exercises
Get help with a step. Ask about MATH 204 tutoring Include the chapter, assessment date and question; attempted work is useful but not required. Sessions are currently in English.

Official course source Your section outline and instructor announcements take priority. These are Nablio study routes and exercises, not official exams.

MATH 204 topics to review

Linear systems and row reductionMatrices, inverses and determinantsDot and cross productsLines, planes and orthogonalityVector spaces and subspacesSpanning sets, independence and basesCoordinates, dimension and linear transformationsEigenvalues, eigenvectors and diagonalization

Common difficulties

Interpreting zero rows, pivots and free variables
Confusing row operations with determinant-preserving operations

Preparing for this course

Use equations, fractions and coordinate geometry confidently. Linear algebra is a separate branch from calculus, not simply the course between Calculus I and II.

Registration requirements

MATH 201 or equivalent is the listed prerequisite. Credit restrictions and section details must be checked with Concordia.

Check the official course catalogue

Try a course problem

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

Show the reasoning

Adding the equations eliminates $y$ : $3x=9$ , so $x=3$ . Substitute into the second equation: $3-y=2$ , giving $y=1$ . Both checks hold: $2(3)+1=7$ and $3-1=2$ .

The line where two planes meet

Use an augmented matrix and a free parameter to find the line where two planes meet. The lesson also compares coincident planes (the same plane) and distinct parallel planes.

Read the full lesson: nine steps, one diagram and three practice questions with explained answers

From a linear transformation to its matrix

Suppose $T$ is a linear transformation from $\mathbb R^2$ to $\mathbb R^2$ with $T(1,0)=(2,1)$ and $T(0,1)=(-1,3)$ . Find $T(3,-2)$ , then find the input whose image is $(1,4)$ .

Build the matrix and check both images

Because $(x,y)=x(1,0)+y(0,1)$ , linearity gives $\begin{aligned}T(x,y)&=x(2,1)+y(-1,3)\\&=(2x-y,\ x+3y).\end{aligned}$ The standard matrix therefore has columns $(2,1)$ and $(-1,3)$ : $A=\begin{pmatrix}2&-1\\1&3\end{pmatrix}.$ These are columns, not rows: multiplying by $(1,0)$ selects the first column.

We obtain $T(3,-2)=(8,-3)$ . For $T(x,y)=(1,4)$ , solve $\begin{aligned}2x-y&=1,\\x+3y&=4.\end{aligned}$ The solution is $(1,1)$ , and substituting gives $T(1,1)=(1,4)$ . Since $\det A=7\ne0$ , each output has exactly one input.

Check a basis image

Use your matrix to recover $T(0,1)$ . Which column should multiplication select?

Check your reasoning

It must recover $(-1,3)$ , the second column. If you obtain $(1,3)$ , check the sign of the first component and make sure the basis images were placed in columns.

How tutoring can help

You can start with a linear-system or vector-geometry question alone. Your tutor can help you write the matrix or draw the geometry, explain the row operations and connect the calculation to the original problem. Comparing cases with one, no or infinitely many solutions can then help you explain what the final matrix means.

Exam preparation for MATH 204

MATH 204 revision after the scheduled midterm

The Fall 2026 outline scheduled the weeks 1–6 midterm for Sunday, November 1, 10:00–11:30 Montréal local time. That scheduled date is now past. It describes a three-hour final covering the whole course, but gives no final date; check the Examinations Office schedule and current instructor announcements. The systems, matrices, inverses, determinants, Cramer’s rule and basic vector work below remain useful early-course practice.

Where to start

Try first: complete “Solve a system and check it” with the answer closed. Then try the question in “Connect multiplication, determinant and inverse”. Keep each row operation visible and substitute your result into the original equations.

Review the step where you got stuck

If a zero row or free variable confuses you, use “Read each final row as an equation”. State whether the whole system is consistent before counting solutions.
If multiplication or inversion is unclear, revisit the matrix–determinant–inverse connection. Add your course’s $3\times3$ inverse, determinant and Cramer exercises; this $2\times2$ example is a starting point.
If a vector answer has the wrong type or length, use dot product and projection. Check norms, distance and orthogonality. The later span/independence section is outside this midterm path.

Mix and retry without notes: solve $x+y=7$ and $x-y=3$ for real x and y by elimination. Use the equations in that order and columns in the order x, y to form the coefficient matrix. Find its determinant, verify the solution, then decide whether the solution vector $(x,y)$ is orthogonal to $(2,-5)$ .

Check the retry

Adding the equations gives $2x=10$ , so $x=5$ and $y=2$ . The coefficient determinant is $1\cdot(-1)-1\cdot1=-2$ , so the solution is unique. Substitution gives $5+2=7$ and $5-2=3$ . The dot product $\begin{aligned}(5,2)\cdot(2,-5)&=10-10\\&=0\end{aligned}$ confirms orthogonality.

You can ask for help with a question even if you have not started it. Bring the question to your instructor, a course tutorial or a tutor; include attempted work if you have some. Work through the first step together, explain the method, then try a similar question with less help.

Flexible routine: use a first attempt to identify one matrix skill and one vector skill to review. Work on them in separate 20–30-minute blocks, then mix questions from the announced exam scope. Explain when elimination, an inverse or Cramer’s rule applies, and check your answer in the original problem.

Fall 2026 outline: assessment p. 2; scope p. 3.

Original Nablio exercises. Adapt the steps to your needs and follow your instructor’s current announcements.

Advice for exams throughout the course (topics vary by exam)

Contrast invertible and singular matrices, independent and dependent vectors, and vectors versus coordinates in a basis. Use your section outline for the scope of eigenvalues and diagonalization.

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.

Independent practice

Free practice and study guides for MATH 204

20 min

Plane Intersections: Row Reduction and Geometry

Find the line where two planes meet using row reduction and a free parameter. Compare intersecting, coincident and parallel planes with a diagram and three explained practice questions.

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.

13 min

Matrices and Vectors: Practice for Concordia MATH 204

Practise systems, row operations, dot products and linear independence with original worked examples and self-checks for Concordia MATH 204.

55 min

Eigenvectors and Eigenvalues: See the Geometry

Find eigenvalues and eigenspaces, then diagonalize a matrix and calculate its powers. Four worked examples and eleven questions with hints and explained solutions.

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MATH 204 · official course reference MATH 204 · official course outline