7.1-01 Invariant directions

Invariant directions

Connect eigenvectors to lines that a linear transformation preserves. Work from geometric language to component checks and small parameter problems, distinguishing a direction from a particular arrow on it.

What you will practise

• Interpret the eigenvector equation
• Check a candidate using matrix multiplication
• Recognize nonzero scalar multiples
• Reason about invariant lines and sums

Prerequisites and study approach

This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review matrix–vector multiplication and scalar multiples. The questions move from recognition and short calculations to connected reasoning. Single-select questions require one answer; multiselect questions ask you to evaluate every statement independently.

A final parameter problem requires the same eigenvalue to satisfy every coordinate. Work out your own result before checking the choices, then use the explanation to compare the reasoning and diagnose a mistaken assumption. This focused practice connects conceptual understanding with reliable problem solving and supports review before the next microlearning.