The Standard Transformations

Rotations, reflections, shears and scalings.

12 problems · about 23 minutes · N-VM.C.11, G-CO.A.2, G-CO.A.5

What this lesson teaches

The student recognises and writes the matrices of common geometric transformations.

Scaling puts factors on the diagonal; rotation uses cosines and sines; shear slides rows sideways.
A reflection has a negative determinant, because it turns the plane over.
Any transformation can be built by multiplying these standard ones together.

Straightforward practice. Get the method working first.

5 problems

A scaling with rows [6, 0] and [0, 2]. What is its determinant?
Answer 12
Why 12.
What does a negative determinant mean?
Answer The transformation has flipped the plane over, reversing orientation.
Why Orientation reversed.
Rows [2, 0] and [0, 5]. What is the determinant?
Answer 10
Why Multiply the diagonal.
A reflection. Is its determinant negative? 1 yes, 0 no.
Answer 1
Why Orientation flips.
A rotation. What is its determinant?
Answer 1
Why Areas are preserved and orientation is kept.

The same ideas with more to keep track of.

3 problems

Rows [1, 7] and [0, 1]. What is the determinant?
Answer 1
Why A shear preserves area.
A rotation by 0 degrees. What is the entry in row 1, column 1?
Answer 1
Why cos 0.
A scaling by 4 in both directions. By what factor does area grow?
Answer 16
Why 4 times 4.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Match each matrix to the transformation it performs.
Answer Rows [1, 0] and [0, −1] → Reflection across the x-axis; Rows [1, 3] and [0, 1] → Shear; Rows [2, 0] and [0, 2] → Uniform scaling
Why Only one of them has a negative determinant.
Rows [0, −1] and [1, 0]. What is the determinant?
Answer 1
Why 0 − (−1).
The Double: A scaling with rows [3, 0] and [0, 3]. What is its determinant?
Answer 9
Why 9.
The Shear: A shear with rows [1, 4] and [0, 1]. What is its determinant?
Answer 1
Why 1.