3.4-05 Outer-product expansion

Outer-product expansion — linear algebra practice

Write a matrix product as a sum of column-row contributions. Compute individual outer products and complete sums, then connect their structure to image directions and dimension bounds. Zero contributions and the role of the shared index are made explicit.

What you will practise

• Compute column-row products
• Assemble outer-product sums
• Infer image-span restrictions

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 multiplication and linear combinations. 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 understand structured and low-rank matrix products.