7.1-03 Reflections and projections

Reflections and projections

Use the geometry of reflections and projections to identify eigenspaces before computing a characteristic polynomial. Move from coordinate-axis examples to planes, oblique projections and a reflection built from a projection.

What you will practise

• Identify fixed and reversed directions
• Find kernels of projections
• Count eigenspace dimensions geometrically
• Connect projection identities to eigenvalues

Prerequisites and study approach

This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review subspaces, perpendicular vectors and basic matrix 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 task connects a projection to a reflection and checks the result on complementary subspaces. 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.