7.2-04 Computing eigenspaces

Computing eigenspaces

Compute eigenspaces as complete null spaces, not just isolated sample vectors. The progression covers diagonal matrices, coupled systems, repeated eigenvalues, row-reduced shifts and a complex eigenvector.

What you will practise

• Solve shifted homogeneous systems
• Choose an independent spanning basis
• Distinguish an eigenspace from its nonzero eigenvectors
• Verify real and complex candidates

Prerequisites and study approach

This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review row reduction, free variables, bases and complex multiplication. 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 problem asks for two full eigenspaces of one three-by-three matrix and checks how their bases fit together. 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.