5.2-02 Subspaces of R³

Connect subspaces of real three-dimensional space to points, lines, planes and the full ambient space. Practice links geometric descriptions to homogeneous equations, parametric forms and the conditions under which a translated set remains linear.

What you will practise

• Classify possible dimensions and geometric types.
• Interpret one or more homogeneous constraints.
• Test whether an affine translate passes through zero.

Preparation and approach

Before starting, review the subspace test and equations of lines and planes. The questions progress from recognition and direct applications to error diagnosis, justification and transfer. Single-select items ask for one defensible answer; multiselect items require you to evaluate every statement independently.

A line or plane is not automatically a subspace; its placement relative to the origin matters. Work through the calculation or argument before checking the explanation, then use the feedback to identify the step or assumption that needs attention.

This microlearning forms part of Module 5, Vector Spaces and the Four Fundamental Subspaces, in the Complete Linear Algebra Course. Use it to connect subspaces of r³ with the surrounding chapter and to revisit the reasoning after an initial attempt.