Matrices as Functions
Unit 4 is not assessed on the AP exam; schools include it based on local requirements, and everything here still builds directly toward calculus and linear algebra. This topic reframes a matrix as a function: input a vector, output the matrix-vector product. The columns record where the basis vectors land, so two columns describe the entire map, and building a matrix means installing the intended basis images as its columns.
The mistakes are function mistakes wearing matrix costumes: multiplying entry-by-entry instead of rows-dot-input, reading the rows as the basis images instead of the columns, composing in the order written instead of right to left (apply A then B is the product BA), and treating the inverse as negation or entrywise reciprocals instead of the undo of the process. The lesson drills each one numerically.
The work
3 ways in · any order
Lesson
Matrices as Functions
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One matrix, one function: vectors in, vectors out. The lesson reads maps off columns, builds matrices from basis images, composes right to left with a worked non-commuting pair, and closes with ten scenarios on products, compositions, identity, inverse, and the linearity contract.
Diagnostic
10-item topic check
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Ten items spanning the two Topic 4.13 misconceptions: entrywise and wrong-order multiplication rules, and misreading what a matrix map does to the plane and its basis vectors. Results route you to the drills that fix what fired.
Targeted Practice
Drill a single misconception
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Pick one of the failure modes you missed and drill it on its own. The round is adaptive: two correct in a row clears the misconception and moves you to the next.