Study Guide

Learning Objectives

• Define linear transformation via constant-multiple output components (4.12.A.1).
• Use L(0)=0 (4.12.A.2).
• Represent vectors as 2x1 or 2xn matrices (4.12.A.3).
• Relate L(v)=Av to a unique 2x2 matrix (4.12.A.4).
• Transform multiple vectors with one matrix multiply (4.12.A.5).

Course scope only (Unit 4 additional topic; not on the AP Exam).

Suggested Practice Sequence

1. Test L(0)=0. 2. Write vectors as column matrices. 3. Compute Av for one vector. 4. Build A from basis images. 5. Batch-transform a 2xn input matrix (4.12.A.5).

Classroom / Check-In Moves

• Does this rule add a constant not tied to x or y?
• Where do i-hat and j-hat go?
• Can we get all outputs in one multiply?

Misconception Watch-List

• Treating translation-only maps as linear (unless embedded in higher dimension).
• Wrong column layout for 2xn inputs.
• Using row vectors when column form is needed for Av.

Vocabulary to Own

• linear transformation; L(v)=Av
• input vector; output vector
• 2x1 column matrix; 2xn batch matrix

Materials Map

• Practice: definition, Av, matrix build, batch outputs
• Word: transformation contexts
• Mixed / multi-*: combine with 4.10 and 4.11
• Review and practice test: transformation fluency
• Interactive: linearity, Av, batch checks