6.1 Orthogonal Subspaces

Module 6 — Orthogonality, Projections and Least Squares. Orthogonal Subspaces. Progressive conceptual, computational and reasoning questions with worked explanations.

by Synapticz Media

Oct 20, 2026

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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.

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6.1-02 Orthogonal complements

Orthogonal complements An orthogonal complement collects every direction perpendicular to a chosen subspace. These questions connect geometric examples with homogeneous equations and dimension counts, while keeping the ambient space explicit. What you will practise Find complements of lines and planes. Solve simultaneous orthogonality constraints. Track dimensions with redundant generators. Reason about inclusions, sums and intersections. 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, homogeneous systems and bases. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports finding residual directions and constructing perpendicular decompositions within Module 6 of the Complete Linear Algebra Course.

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6.1-03 Row space and null space

Row space and null space Explore the relationship between the null space and row space of a real matrix. This set develops arguments from row-by-vector multiplication, then uses them to interpret kernels, rank and the action of a matrix on its domain. What you will practise Read a zero output as row orthogonality. Prove both directions of the row–null relationship. Recover null directions from row-reduced equations. Explain injectivity on the row space. 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 row reduction, rank–nullity and matrix–vector products. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports the input-side half of the complete four-subspace picture within Module 6 of the Complete Linear Algebra Course.

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6.1-04 Column space and left null space

Column space and left null space Column space and left null space describe complementary parts of the output space. Practice moves from the transpose equation to concrete reachability tests and certificates that a target cannot be produced by a matrix. What you will practise Identify the left null space and its ambient dimension. Compute perpendicular output directions. Test targets against compatibility conditions. Explain why a left-null vector can certify inconsistency. 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 transpose multiplication, column spans and homogeneous systems. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports understanding the location of least-squares residuals within Module 6 of the Complete Linear Algebra Course.

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6.1-05 Unique orthogonal decomposition

Unique orthogonal decomposition Unique orthogonal decomposition separates a vector into a component in a subspace and a perpendicular remainder. The questions connect explicit decompositions with the proof of uniqueness, energy identities and the nearest-point interpretation. What you will practise Compute coordinate and noncoordinate decompositions. Check every condition of a proposed split. Use squared-length identities. Distinguish orthogonal decomposition from a general direct sum. 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 complements, dot products and basic subspace arguments. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports projection matrices and closest-point problems within Module 6 of the Complete Linear Algebra Course.

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6.1-06 The complete four-subspace picture

The complete four-subspace picture Bring the four fundamental subspaces into one coherent account of a linear map. This questionset emphasizes which spaces belong to the domain, which belong to the codomain, and how rank and orthogonality connect them. What you will practise Locate and count all four subspaces. Explain what happens to a null-space component. Describe the map from row space to column space. Interpret minimum-norm solutions through perpendicular components. 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 the row–null and column–left-null relationships. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports least squares, inverse problems and later singular-value analysis within Module 6 of the Complete Linear Algebra Course.

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