What quantity did you change?
Labs / Mathematics
Matrix Transformations
A matrix is just an instruction for moving space. Drag the sliders for the four entries a, b, c & d and watch the grid, the basis vectors î and ĵ, and the F shape bend along with them. Grab the F to slide it around, and read off the determinant — the factor by which every area is scaled.
Labs / Mathematics / Matrix Transformations
Matrix Transformations
A matrix is just an instruction for moving space. Drag the sliders for the four entries a, b, c & d and watch the grid, the basis vectors î and ĵ, and the F shape bend along with them. Grab the F to slide it around, and read off the determinant — the factor by which every area is scaled.
Why it matters
Why explore this?
Mathematics helps us describe patterns precisely and predict what comes next. This model turns an abstract relationship into something you can change and observe.
Explore
Try the model
M [1.00 1.00; 0.00 1.00]det 1.00invertible yes
Investigate
Questions to test
Predict first, try it in the model, then use what you observe to answer.
What to try
- Where do î and ĵ end up? Notice that they are always exactly the two columns of the matrix.
- Hit Shear and check the determinant. Why does the F look slanted yet still cover the same area?
- Drive the determinant to zero. What happens to the whole grid, and why does “invertible” flip to no?
- Make the determinant negative with Reflect. How can you tell the plane has been flipped over rather than just turned?
- Combine a rotation with a stretch by hand — can you build a transformation that turns the F upside down?
Explain
Make sense of what happened
State the pattern in words, test it with another example, and then connect it to the mathematical rule or representation shown in the model.
Check
Before you leave
What pattern did you observe?
Does the pattern work for another example?
If you can answer all three using evidence from the model, you understand the main idea. If not, return to Explore and change one setting at a time.