University · Linear algebra
Watch a matrix reshape the plane
Stretch and shear a unit square, and compare its area with the determinant.
Predict what will happen, press Play, then pause and explain what changed. Every control also works without playback.
Watch the relationship
PausedHD animation studio
From experiment to screen.
Present a crisp Canvas scene, save a full-HD image, or capture your model as a silent video.
Understand what you are seeing
The idea behind the motion.
A linear map is determined by where it sends the two coordinate basis vectors. The columns of its matrix are those destinations. A unit square becomes a parallelogram, and the magnitude of the determinant tells you its area.
A relationship to keep
A = [[sₓ, k], [0, sᵧ]]; det(A) = sₓsᵧ
Read the symbols alongside the explanation. A diagram shows the relationship; the assumptions tell you when it applies.
- STEP 1
Start with a unit square
Its sides are e₁ = (1, 0) and e₂ = (0, 1), so the original area is one square unit. The faint square stays in place for comparison.
- STEP 2
Follow the columns
The first side moves toward (sₓ, 0); the second moves toward (k, sᵧ). All points use the same linear map, so parallel grid lines remain parallel.
- STEP 3
Measure the area
At the end, the parallelogram has base sₓ and perpendicular height sᵧ. Shearing changes its slant but does not change that height or its area.
Your turn to explain
Make a prediction. Test your reasoning.
What is the final area when sₓ = 2, sᵧ = 1.5, and k = −1?
Use the values specified in the question. Reset restores the initial values for this investigation.
Compare your explanation
3 square units. The determinant is 2 × 1.5 = 3; the shear does not enter the determinant for this triangular matrix.
Work through a full lesson
Connect the animation to a worked example and practice questions.