6.1-02 Orthogonal complements

Orthogonal complements

An orthogonal complement collects every direction perpendicular to a chosen subspace. These questions connect geometric examples with homogeneous equations and dimension counts, while keeping the ambient space explicit.

What you will practise

• Find complements of lines and planes.
• Solve simultaneous orthogonality constraints.
• Track dimensions with redundant generators.
• Reason about inclusions, sums and intersections.

Begin with recognition and interpretation, then work through calculations, error diagnosis and more demanding reasoning. Single-select and multi-select questions ask you to evaluate the stated assumptions as well as the result. Worked explanations connect the answer to a method or a counterexample, so you can check why tempting alternatives fail.

Before you start

Review dot products, homogeneous systems and bases. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports finding residual directions and constructing perpendicular decompositions within Module 6 of the Complete Linear Algebra Course.