7.4-01 Constructing an eigenbasis

Constructing an eigenbasis

Build an eigenbasis by combining independent directions from the relevant eigenspaces. The set emphasizes spanning the entire space, repeated eigenvalues and the choice between real and complex scalars.

What you will practise

• Test independence of eigenvectors
• Assemble bases from distinct eigenspaces
• Use dimension counts to detect a deficit
• Specify the field for an eigenbasis

Prerequisites and study approach

This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review bases, eigenspaces and determinants. 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 selection task compares several candidate bases for a matrix with a repeated eigenvalue. 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.