6.3-05 When the Gram matrix is invertible

When the Gram matrix is invertible

Investigate when a Gram matrix can be inverted and how this relates to its generating columns. This questionset develops a squared-norm proof, checks rectangular examples and connects rank, positivity and uniqueness.

What you will practise

• Relate the Gram quadratic form to output length.
• Prove null-space equality.
• Test full-column-rank requirements.
• Analyze parameters and impossible wide-matrix cases.

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 null spaces, rank and matrix transposes. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports knowing when inverse-Gram projection formulas are valid within Module 6 of the Complete Linear Algebra Course.