7.4-04 Algebraic and geometric multiplicity

Algebraic and geometric multiplicity

Compare algebraic multiplicity, which comes from a characteristic polynomial, with geometric multiplicity, which comes from a null space. Use scalar matrices, defective blocks and rank data to see both equality and deficiency.

What you will practise

• Read algebraic multiplicity
• Compute geometric multiplicity
• Apply multiplicity bounds
• Count the available independent eigenvectors

Prerequisites and study approach

This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review polynomial factors, kernels and rank-nullity. 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 last examples combine repeated-root data with shifted ranks to decide whether enough independent eigenvectors exist. 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.