synapticz

6.3 Orthogonal Projection onto a Subspace

Module 6 — Orthogonality, Projections and Least Squares. Orthogonal Projection onto a Subspace. Progressive conceptual, computational and reasoning questions with worked explanations.

by Synapticz Media
Oct 20, 2026
8 quizzes
Public
FREE

Explore this collection

Browse the topics and quizzes below. Expand or collapse any branch to focus on a subject.

Quiz Book Content

8 quizzes

Showing 1–8 of 8 quizzes · Page 1 of 1

6.3-01 A line projection as a matrix

6.3-01 A line projection as a matrix

A line projection as a matrix Projection onto a line is the smallest complete model of orthogonal approximation. Work from a scalar orthogonality equation to the outer-product matrix, then check scaling invariance, signs, distances and the range of the result. What you will practise Derive the projection coefficient. Construct and apply line-projector matrices. Separate projection from residual. Use squared lengths to calculate distance. 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, norms and outer products. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports projection onto higher-dimensional subspaces within Module 6 of the Complete Linear Algebra Course.

linear algebra
FREE
Start Quiz
6.3-02 Idempotence and symmetry

6.3-02 Idempotence and symmetry

Idempotence and symmetry Idempotence and symmetry explain the algebra of orthogonal projection. The questions distinguish those two properties from invertibility and orthogonality of a square matrix, using concrete tests and short structural arguments. What you will practise Interpret repeated projection. Check symmetry and idempotence separately. Use range, kernel and energy identities. Investigate allowable eigenvalues and boundary 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 matrix multiplication and orthogonal decomposition. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports recognizing projectors and distinguishing oblique projection within Module 6 of the Complete Linear Algebra Course.

linear algebra
FREE
Start Quiz
6.3-03 Residual orthogonality

6.3-03 Residual orthogonality

Residual orthogonality Residual orthogonality is the geometric condition behind least squares. This set connects the nearest point in a column space with matrix equations, directional changes in the objective and a proof of global optimality. What you will practise Locate residuals in observation space. Check perpendicularity to every design column. Derive the first-order condition by expansion. Distinguish sufficient tests from weaker one-vector checks. 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 subspace projection and matrix–vector multiplication. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports deriving and verifying the normal equations within Module 6 of the Complete Linear Algebra Course.

linear algebra
FREE
Start Quiz
6.3-04 Normal equations

6.3-04 Normal equations

Normal equations The normal equations convert a closest-point problem into a linear system for coefficients. Practice includes building Gram matrices, forming right-hand sides, solving small systems and interpreting solutions when columns are dependent. What you will practise Form a dimensionally consistent normal system. Solve one- and two-parameter fits. Calculate residual error and minimizer families. Compare exact algebra with numerical method 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 residual orthogonality and solving linear systems. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports data fitting and the QR alternative to Gram-matrix computation within Module 6 of the Complete Linear Algebra Course.

linear algebra
FREE
Start Quiz
6.3-05 When the Gram matrix is invertible

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.

linear algebra
FREE
Start Quiz
6.3-06 The general projection matrix

6.3-06 The general projection matrix

The general projection matrix General subspace projection can be computed from any independent column basis. These questions follow the inverse-Gram formula, simplify it for orthonormal columns and test how changing the basis affects the resulting closest point. What you will practise Construct correctly sized projection matrices. Project onto coordinate and noncoordinate planes. Compare different bases for the same span. Handle redundant columns without using an invalid inverse. 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 normal equations and Gram-matrix invertibility. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports basis-independent interpretation of least-squares fitted values within Module 6 of the Complete Linear Algebra Course.

linear algebra
FREE
Start Quiz
6.3-07 Complementary projections

6.3-07 Complementary projections

Complementary projections Complementary projectors split a vector into its fitted component and perpendicular residual. Practice connects the identity-minus-projector construction with left null spaces, dimension counts, energy calculations and reflections. What you will practise Construct and apply a complementary projector. Locate its range and kernel. Check mutual annihilation and reconstruction. Relate complementary components to reflection. 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 idempotence, symmetry and orthogonal decomposition. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports interpreting residuals and Householder geometry within Module 6 of the Complete Linear Algebra Course.

linear algebra
FREE
Start Quiz
6.3-08 Recognizing orthogonal projectors

6.3-08 Recognizing orthogonal projectors

Recognizing orthogonal projectors An algebraic projection need not be a closest-point projection. Through symmetric-idempotent tests and explicit oblique examples, this set explains why orthogonality of range and kernel is the decisive extra condition. What you will practise Recognize orthogonal projectors. Compute range and kernel of oblique projectors. Compare projected points with nearest points. Prove the symmetric-idempotent characterization. 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 projection matrices and range–kernel decomposition. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports evaluating projection claims without confusing them with general idempotence within Module 6 of the Complete Linear Algebra Course.

linear algebra
FREE
Start Quiz