Work with 2×2 Matrices as Transformations

Learning objectives:

• Describe how a matrix transforms the plane by tracking basis vectors
• Sketch the image of the unit square under a given transformation
• Interpret as the area scaling factor
• Explain what the sign of tells you about orientation
• Connect to non-invertibility geometrically

Every Game Effect Is a Matrix

A matrix transforms every pixel's coordinates at once:

• Scale: doubles the size
• Reflect: flips left/right
• Rotate: spins 90° CCW
• Shear: slides one axis

One question spans all of these: what does the matrix do to area?

Track Two Vectors to Know Everything

Key bridge: Applying to and determines what does to every vector.

For :

So and tell you the whole story.

Those two results are exactly the columns of .

Columns Are Images of Basis Vectors

For :

To understand a transformation: apply to and — read off the columns.

Unit Square to Parallelogram: Shear

• : ,

Unit Square: Rotation and Reflection

Rotation 90° CCW:

• ,
• Image: unit square rotated — still area 1

Reflection over -axis:

• ,
• Image: unit square reflected — still area 1

Quick Check: Find the Image Vertices

For each matrix, find the four vertices of the image unit square:

1. — what shape is the image?

2. — what shape is the image?

Hint: apply to , , , and .

The Parallelogram Has a Measurable Area

The image parallelogram is spanned by columns and .

Area of a 2D parallelogram from two vectors:

The determinant is the area of the image parallelogram when the input is the unit square.

Area Check: Shear, Rotation, Reflection

Shear : → area

Rotation : → area

Reflection : → , area

All three checks confirm: = area of the image parallelogram.

Sign of the Determinant: Orientation

• : orientation preserved — CCW stays CCW
• : orientation reversed — CCW becomes CW
• : degenerate — area collapses to 0

Orientation Reversal Is Not Just Flipping

Reflection over : ,

Image is reflected across the diagonal — not "upside down."

Rule: Any reflection over any line through the origin has .

Orientation reversal = CCW flips to CW, not vertical flipping.

Classify Each Transformation by Its Properties

Compute , state the area factor, orientation, and invertibility:

1. — uniform scaling by 3

2. — horizontal shear

3. — rotation 90° CW

4. — is this invertible?

Degenerate Case: When the Determinant Is Zero

:

• Columns are parallel: — image degenerates to a line

Why det = 0 Means Not Invertible

Geometric argument: when , distinct inputs map to the same output.

For : both and map to .

No inverse can exist: given output , which input was it?

VM.C.10 theorem: is invertible — now with geometric meaning.

Inverse Matrix Reverses the Whole Transformation

(90° CCW). What is ?

Using the inverse formula with :

Geometric check: is the 90° CW rotation.

rotates backward — it reverses the transformation.

Unshearing: Inverse of a Shear

(horizontal shear by )

— always invertible.

Verify: ✓

Write the Full Transformation Story

"Apply a horizontal shear by 2, then scale by 3."

1. Write matrices (shear) and (scale)
2. Write the composite — which goes on the right?
3. Find the image of the unit square
4. Compute

Answer: area? invertible? orientation preserved?

Spot the Determinant Size Misconception

A student says: "Large entries → large determinant."

:

Columns are parallel (column 2 = column 1) — det = 0.

The determinant measures column relationships, not entry magnitudes.

What a 2×2 Matrix Encodes

The VM Cluster in One Picture

A matrix moves the entire plane.

• C.6–C.8: add, scale, multiply
• C.9: order matters
• C.10: det checks invertibility
• C.11: transforms one vector
• C.12: transforms every point; = area

Forward: eigenvalues, 3D volume, the full theory of linear maps.