7.1-05 Shears and missing eigen-directions

Shears and missing eigen-directions

Explore how a shear can be invertible while providing too few eigen-directions for a basis. Coordinate calculations connect the familiar slanted-grid picture to a repeated eigenvalue and a deficient eigenspace.

What you will practise

• Find the fixed line of a shear
• Solve the shifted homogeneous equation
• Compare a nontrivial shear with the identity
• Track repeated shear applications

Prerequisites and study approach

This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review coordinate transformations and elementary homogeneous systems. 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 task links the missing independent direction to failure of diagonalization. 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.