6.2-05 Rectangular orthonormal-column matrices

Rectangular orthonormal-column matrices

Tall matrices can have orthonormal columns without being square orthogonal matrices. This microlearning follows the two multiplication orders carefully and explains embedding, coordinate extraction, projections and discarded perpendicular directions.

What you will practise

• Track dimensions of both Gram products.
• Test input-side length preservation.
• Identify fixed and discarded output directions.
• Complete an orthonormal family and interpret one-sided inverses.

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 orthonormal coordinates and matrix dimensions. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports reduced QR and subspace projection formulas within Module 6 of the Complete Linear Algebra Course.