3.4-02 Composition order and dimensions

Composition order and dimensions — linear algebra practice

Read matrix products as successive transformations of column vectors. Track the input, intermediate and output dimensions through two- and three-stage pipelines. Concrete shear and scaling examples show why both compatibility and application order matter.

What you will practise

• Check inner dimensions
• Determine product and input sizes
• Translate pipelines into ordered products

This microlearning belongs to Module 3 of the Complete Linear Algebra Course. Single-select and multiselect questions progress from interpretation to calculation, application and error diagnosis. Each problem is self-contained, and the explanations show how to justify a result rather than only identify a choice. In multiselect tasks, assess each statement separately; the number of correct choices varies.

Before starting, review matrix-vector multiplication and map domains. Work through the reasoning on paper, compare your method with the explanation and revisit the assumptions behind any missed item. These skills prepare you to compute matrix products with correct interpretation.