7.5-02 Real bases from a complex eigenvector

Real bases from a complex eigenvector

Turn a complex eigenvector into a real invariant plane using its real and imaginary parts. Derive their coupled images and distinguish coordinate rotation–scaling from Euclidean behavior in a nonorthogonal basis.

What you will practise

• Extract real and imaginary vectors
• Derive their image formulas
• Prove real independence
• Identify the invariant plane

Prerequisites and study approach

This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review complex multiplication, real spans and basis changes. 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.

The final interpretation question checks what is preserved by similarity and what requires an orthonormal basis. 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.