6.5 Gram–Schmidt and QR

Module 6 — Orthogonality, Projections and Least Squares. Gram–Schmidt and QR. Progressive conceptual, computational and reasoning questions with worked explanations.

by Synapticz Media

Oct 20, 2026

8 quizzes

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8 quizzes

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6.5-01 One orthogonalization step

One orthogonalization step One orthogonalization step removes the part of a vector already represented by earlier orthonormal directions. Practice follows the subtraction carefully, checks the remaining direction and handles normalization and nonunit basis vectors. What you will practise Subtract one or several orthogonal components. Calculate residuals and normalize them. Explain why the partial span is preserved. Handle already perpendicular and zero-residual 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 orthogonal projection and normalization. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports the complete Gram–Schmidt algorithm within Module 6 of the Complete Linear Algebra Course.

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6.5-02 The Gram–Schmidt algorithm

The Gram–Schmidt algorithm Construct orthonormal bases using the full Gram–Schmidt process. The questions build from two-vector calculations to three-dimensional examples, partial-span invariants, order dependence and the inner-product reasoning behind the algorithm. What you will practise Carry out successive projection subtractions. Normalize each independent residual. Track output order and partial spans. Diagnose incorrect subtraction against overlapping inputs. 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 one-step orthogonalization and linear independence. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports QR factorization and orthogonal bases for abstract spaces within Module 6 of the Complete Linear Algebra Course.

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6.5-03 Dependence during Gram–Schmidt

Dependence during Gram–Schmidt A zero Gram–Schmidt residual means a direction was already represented. This set distinguishes exact redundancy from a small floating-point residual and explains how to continue constructing a basis without inventing new directions. What you will practise Recognize dependent inputs and count retained directions. Skip exact zero residuals correctly. Analyze a near-dependent parameter example. Use scale-aware language for numerical rank decisions. 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 Gram–Schmidt and rank. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports rank-aware orthogonalization and numerical methods within Module 6 of the Complete Linear Algebra Course.

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6.5-04 Constructing QR

Constructing QR QR factorization records Gram–Schmidt coordinates in an upper triangular matrix. Practice covers reduced factor shapes, coefficient entries, reconstruction, diagonal signs, column spans and the convention that makes the factors unique. What you will practise Construct and verify reduced QR factors. Explain why R is upper triangular. Recover columns and basis coordinates. Handle sign changes and positive-diagonal conventions. 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 Gram–Schmidt and matrix products. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports triangular least-squares solves within Module 6 of the Complete Linear Algebra Course.

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6.5-05 Least squares with QR

Least squares with QR Solve least-squares problems using orthonormal coordinates and a triangular system. This set follows reduced and full QR viewpoints, computes fitted vectors and residual errors, and identifies the rank assumptions required for a unique solve. What you will practise Form the QR least-squares right-hand side. Solve R by back substitution. Recover fitted outputs and residual norms. Compare QR with exact normal-equation solutions. 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 reduced QR and ordinary least squares. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports solving fits without explicitly forming the Gram matrix within Module 6 of the Complete Linear Algebra Course.

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6.5-06 Modified Gram–Schmidt

Modified Gram–Schmidt Modified Gram–Schmidt changes when projection coefficients are computed. These questions trace the updated residual, explain exact-arithmetic equivalence with the classical form and examine the numerical purpose of reorthogonalization. What you will practise Compare original-column and updated-residual calculations. Execute a modified orthogonalization step. Explain rounding and cancellation effects. Use reconstruction and orthogonality diagnostics. 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 classical Gram–Schmidt and basic floating-point awareness. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports judging orthogonalization quality in numerical work within Module 6 of the Complete Linear Algebra Course.

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6.5-07 Householder reflections

Householder reflections Householder reflections eliminate components while preserving geometric information. The set develops the rank-one formula, tests fixed and reversed directions, performs a numerical example and explains reflection ordering and stable sign choices. What you will practise Construct and apply a Householder reflector. Distinguish reflection from projection. Use reflectors to introduce zeros. Track products and numerical cancellation choices. 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 orthogonal projectors, matrices and norms. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports Householder QR and stable matrix factorization within Module 6 of the Complete Linear Algebra Course.

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6.5-08 Comparing numerical methods

Comparing numerical methods Numerical QR methods can agree mathematically while differing in rounding behavior. These questions compare classical and modified Gram–Schmidt with Householder QR, separating conditioning of the problem from stability of the algorithm. What you will practise Interpret exact versus floating-point equivalence. Recognize the cost of forming a Gram matrix. Assess both orthogonality and reconstruction errors. Explain why stable methods do not remove intrinsic sensitivity. 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 QR, normal equations and numerical near-dependence. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports choosing and evaluating numerical linear-algebra methods within Module 6 of the Complete Linear Algebra Course.

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