Work with 2x2 matrices as transformations

What you'll learn

1. Describe how a $2 \times 2$ matrix transforms the plane by tracking where the standard basis vectors go.
2. Sketch the image of the unit square under a given $2 \times 2$ transformation.
3. Interpret $|\det(A)|$ as the factor by which $A$ scales areas.
4. Explain the geometric meaning of the sign of $\det(A)$: positive = orientation preserved; negative = orientation reversed.
5. Explain why $\det(A) = 0$ corresponds to a non-invertible (area-collapsing) transformation and connect this to VM.C.10.

Prerequisites

• Understand zero and identity matrices
• Multiply matrices and vectors