Family Overview
In Plain Language
Topic 4.12 connects linear transformations to 2x2 matrices.
- Linear transformation: L maps input vectors to output vectors; each output component is a sum of constant multiples of input components (4.12.A.1).
- Zero vector: L(0)=0 (4.12.A.2).
- Matrix form: one vector is a 2x1 matrix; n vectors can be stored as a 2xn matrix (4.12.A.3).
- Key link: for each linear L on R^2, there is a unique 2x2 matrix A with L(v)=Av; conversely Av defines a linear transformation (4.12.A.4).
- Batch form: A times a 2xn input matrix gives all n outputs at once (4.12.A.5).
Course scope only (Unit 4 additional topic; not on the AP Exam).
Why It Matters
Matrix transformations prepare for 4.13 (matrices as functions) and geometric models in 4.14.
How Families Can Help
- Ask what happens to the zero vector first.
- Build A from where i-hat and j-hat go.
- Check outputs with matrix multiply from 4.10.
What Not to Do
- Do not treat nonlinear rules (with constants added to inputs squared, etc.) as linear.
- Do not skip the L(0)=0 check.
- Do not confuse a single vector with a 2xn batch matrix layout.