7.5-03 The real two-by-two block

The real two-by-two block

Construct the real two-by-two rotation–scaling block with signs tied to an explicitly ordered basis. Work through basis reflections, coordinate orientation, modulus calculations and verification of a concrete eigenpair.

What you will practise

• Build a matrix from image columns
• Track off-diagonal signs
• Calculate scale and rotation direction
• Distinguish basis changes from changed spectra

Prerequisites and study approach

This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review complex eigenpairs and coordinates in a real basis. 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 example checks a consequence of similarity that survives even when standard-coordinate angles and lengths do not. 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.