3.3-03 Dimension constraints

Dimension constraints — linear algebra practice

Use row and column counts to determine possible injectivity and surjectivity behavior. Compare wide, tall and square matrices, count free variables and separate necessary inequalities from guarantees. The questions require reasoning about structure before choosing specific coefficients.

What you will practise

• Apply pivot-count bounds
• Classify rectangular maps
• Distinguish necessary and sufficient dimension conditions

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 injectivity, surjectivity and free variables. 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 design maps with feasible prescribed behavior.