6.1-01 Orthogonality of entire subspaces

Orthogonality of entire subspaces

Orthogonality of entire subspaces is stronger than finding one perpendicular pair or observing that two spaces meet only at zero. This microlearning develops the basis-level tests that make a universal geometric claim manageable.

What you will practise

• Interpret orthogonality between two spans.
• Check all cross-basis dot products.
• Use dimension restrictions and parameter conditions.
• Distinguish direct sums from orthogonal sums.

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, spans, bases and subspace dimension. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports orthogonal complements and decompositions of the four fundamental subspaces within Module 6 of the Complete Linear Algebra Course.