3.3-01 Injectivity and the zero preimage

Injectivity and the zero preimage — linear algebra practice

Relate distinct-input preservation to the vectors that a linear map sends to zero. Prove the criterion using an input difference, solve concrete homogeneous equations and test a parameter family. A nonlinear counterexample clarifies the hypothesis needed for the familiar shortcut.

What you will practise

• Use the zero-preimage criterion
• Connect injectivity to independent columns
• Test parameters and the limits of the criterion

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 homogeneous systems, pivots and linearity. 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 recognize when outputs retain all input information.