7.4-03 Independence of distinct-eigenvalue vectors

Independence of distinct-eigenvalue vectors

Understand why eigenvectors associated with distinct eigenvalues are independent, and what this theorem does not say. Computation, counterexamples and short proof steps connect the theorem to diagonalization.

What you will practise

• Apply the independence theorem
• Separate independence from orthogonality
• Isolate coefficients using shifted operators
• Recognize the limits of the converse

Prerequisites and study approach

This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review linear dependence and eigenvector equations. 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 reasoning problem examines sums across distinct eigenspaces and explains why they normally stop being eigenvectors. 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.