Linear algebra in CEGEP: understand it instead of memorizing it

Why linear algebra throws so many students

Linear Algebra and Vector Geometry (201-NYC) is part of the CEGEP Science program. At Cégep à distance, its prerequisite is Secondary 5 TS or SN math. The course generally covers systems of equations, matrices, determinants, vectors, lines and planes.

Many students treat it as a list of recipes: run the algorithm, get a number. That works until the first question that asks what a result means. The good news is that almost every computation in the course has a simple geometric meaning. Once you see it, there's far less to memorize.

Gauss-Jordan elimination, step by step

Let's solve x + y + z = 6, 2x − y + z = 3, x + 2y − z = 2. Write the augmented matrix, one row per equation: R1 = [1 1 1 | 6], R2 = [2 −1 1 | 3], R3 = [1 2 −1 | 2]. The goal is 1s on the diagonal and 0s everywhere else.

1. R2 ← R2 − 2R1 gives [0 −3 −1 | −9].
2. R3 ← R3 − R1 gives [0 1 −2 | −4].
3. Swap R2 and R3 so the pivot is 1: R2 = [0 1 −2 | −4], R3 = [0 −3 −1 | −9].
4. R3 ← R3 + 3R2 gives [0 0 −7 | −21].
5. R3 ← (−1/7)R3 gives [0 0 1 | 3].
6. R2 ← R2 + 2R3 gives [0 1 0 | 2].
7. R1 ← R1 − R2 − R3 gives [1 0 0 | 1].

Solution: x = 1, y = 2, z = 3. Check all three equations: 1 + 2 + 3 = 6, 2 − 2 + 3 = 3 and 1 + 4 − 3 = 2. Geometrically, each equation is a plane in space, and the solution is the point where all three planes meet. A row like [0 0 0 | 5] means the system is inconsistent (no solution), while a row of [0 0 0 | 0] leaves a free variable and infinitely many solutions.

What a determinant actually measures

A determinant is more than a formula: it's an area or volume scaling factor. For the 2 × 2 matrix with rows [3 1] and [2 4], det = 3 × 4 − 1 × 2 = 10. The columns (3, 2) and (1, 4) span a parallelogram of area 10, and the matching transformation multiplies every area by 10. In 3 × 3, |det| is the volume of the parallelepiped spanned by the three columns.

• The sign tells you whether orientation is preserved or flipped.
• A zero determinant means space gets squashed into a lower dimension: the matrix isn't invertible and the system has no unique solution.
• The matrix of the system above has determinant 7, which is nonzero and confirms the unique solution.

Row operations affect it predictably: swapping two rows flips the sign, multiplying a row by k multiplies the determinant by k, and adding a multiple of one row to another leaves it unchanged. Also, det(AB) = det(A) det(B).

Dot product and cross product

Take u = (1, 2, 3) and v = (4, −5, 6).

Dot product: a number

u · v = 1(4) + 2(−5) + 3(6) = 12. It measures how much two vectors point the same way, since u · v = ||u|| ||v|| cos θ. Here ||u|| = √14 and ||v|| = √77, so cos θ = 12/√1078 and θ ≈ 68.6°. A dot product of zero means the vectors are perpendicular.

Cross product: a vector

u × v = (2 × 6 − 3 × (−5), 3 × 4 − 1 × 6, 1 × (−5) − 2 × 4) = (27, 6, −13). This vector is perpendicular to both u and v, which you can confirm with dot products: 27 + 12 − 39 = 0 and 108 − 30 − 78 = 0. Its length, √934 ≈ 30.6, is the area of the parallelogram spanned by u and v.

Lines and planes in space

Keep two ideas in mind: a line is described by a point and a direction vector, a plane by a point and a normal vector.

• Plane from a point and a normal. With P(1, 0, 2) and n = (2, −1, 3): 2(x − 1) − (y − 0) + 3(z − 2) = 0, which simplifies to 2x − y + 3z = 8.
• Plane through three points. For A(1, 0, 0), B(0, 1, 0) and C(0, 0, 1), compute AB × AC = (−1, 1, 0) × (−1, 0, 1) = (1, 1, 1), so the plane is x + y + z = 1.
• Line meets plane. The line (x, y, z) = t(1, 1, 1) hits the plane 2x − y + 3z = 8 when 2t − t + 3t = 8, so t = 2, at the point (2, 2, 2).
• Distance from a point to a plane. |ax0 + by0 + cz0 − d| / √(a^2 + b^2 + c^2). The origin is 1/√3 away from x + y + z = 1.

A line is perpendicular to a plane when its direction vector is parallel to the normal, and parallel to the plane when their dot product is zero.

The most common mistakes

• Assuming AB = BA. Matrix multiplication isn't commutative, and (AB)^−1 = B^−1 A^−1, in that order.
• Writing det(A + B) = det(A) + det(B). It's false in general. And for a 3 × 3 matrix, det(2A) = 8 det(A), not 2 det(A).
• Forgetting the augmented column during a row operation, or doing two operations at once using a row you've already changed.
• Dividing by an expression that could be zero in a system with a parameter, which loses cases.
• Reversing a cross product. v × u = −(u × v), and the cross product only exists in three dimensions.
• Mixing up direction and normal vectors when moving between lines and planes.

A study method built on understanding

• Draw. Sketch the vectors, planes and parallelograms. A quick picture prevents a lot of sign errors.
• Always check. Plug your solution back into the original system, and test a cross product for perpendicularity with two dot products.
• Explain each result in one sentence. "The determinant is zero, so the three vectors lie in the same plane." If you can't, the idea hasn't clicked yet.
• Use more than one source. Gilbert Strang's video lectures on MIT OpenCourseWare give an excellent geometric perspective.

For one-on-one help, see our CEGEP linear algebra tutoring page or book a session with Stellaire Académie. If you review with AI tools, our guide on studying with AI without cheating can help too.

Frequently asked questions