5.2 Subspaces and Their Combinations

Module 5 — Vector Spaces and the Four Fundamental Subspaces. Subspaces and Their Combinations. Practice connects definitions, worked calculations, counterexamples and structural reasoning.

by Synapticz Media

Oct 20, 2026

7 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

7 quizzes

Showing 1–7 of 7 quizzes · Page 1 of 1

5.2-01 The subspace test

Apply the subspace test to equations, inequalities, parametric descriptions and geometric sets. The questions move from detecting a missing zero to justifying closure and diagnosing cases that pass one part of the test but fail another. What you will practise Check zero, addition and real scalar closure. Rewrite a parametric set as a span. Choose explicit counterexamples for nonlinear or one-sided constraints. Preparation and approach Before starting, review vector-space operations and basic coordinate geometry. The questions progress from recognition and direct applications to error diagnosis, justification and transfer. Single-select items ask for one defensible answer; multiselect items require you to evaluate every statement independently. Containing zero and being closed under one operation are not enough to establish a subspace. Work through the calculation or argument before checking the explanation, then use the feedback to identify the step or assumption that needs attention. This microlearning forms part of Module 5, Vector Spaces and the Four Fundamental Subspaces, in the Complete Linear Algebra Course. Use it to connect the subspace test with the surrounding chapter and to revisit the reasoning after an initial attempt.

linear algebra

FREE

Start Quiz

5.2-02 Subspaces of R³

Connect subspaces of real three-dimensional space to points, lines, planes and the full ambient space. Practice links geometric descriptions to homogeneous equations, parametric forms and the conditions under which a translated set remains linear. What you will practise Classify possible dimensions and geometric types. Interpret one or more homogeneous constraints. Test whether an affine translate passes through zero. Preparation and approach Before starting, review the subspace test and equations of lines and planes. The questions progress from recognition and direct applications to error diagnosis, justification and transfer. Single-select items ask for one defensible answer; multiselect items require you to evaluate every statement independently. A line or plane is not automatically a subspace; its placement relative to the origin matters. Work through the calculation or argument before checking the explanation, then use the feedback to identify the step or assumption that needs attention. This microlearning forms part of Module 5, Vector Spaces and the Four Fundamental Subspaces, in the Complete Linear Algebra Course. Use it to connect subspaces of r³ with the surrounding chapter and to revisit the reasoning after an initial attempt.

linear algebra

FREE

Start Quiz

5.2-03 Abstract subspace examples

Apply subspace reasoning to polynomials, matrices and functions. The set emphasizes homogeneous linear constraints, then contrasts them with singularity, fixed values and nonlinear conditions that can fail even when they look algebraically natural. What you will practise Test coefficient, trace and symmetry conditions. Construct counterexamples to nonlinear matrix constraints. Express constrained polynomials using free parameters. Preparation and approach Before starting, review the subspace test, polynomial coefficients and basic matrix operations. The questions progress from recognition and direct applications to error diagnosis, justification and transfer. Single-select items ask for one defensible answer; multiselect items require you to evaluate every statement independently. A condition involving a familiar matrix quantity is not necessarily a linear constraint. Work through the calculation or argument before checking the explanation, then use the feedback to identify the step or assumption that needs attention. This microlearning forms part of Module 5, Vector Spaces and the Four Fundamental Subspaces, in the Complete Linear Algebra Course. Use it to connect abstract subspace examples with the surrounding chapter and to revisit the reasoning after an initial attempt.

linear algebra

FREE

Start Quiz

5.2-04 Intersections and unions

Understand how intersections and unions behave differently for subspaces. Geometric examples lead into closure proofs, containment conditions and the key subtraction argument used to diagnose a union that does not stay closed under addition. What you will practise Compute intersections of coordinate planes. Distinguish nested unions from nonnested unions. Explain closure for an arbitrary intersection of subspaces. Preparation and approach Before starting, review set membership, subspace closure and elementary geometry. The questions progress from recognition and direct applications to error diagnosis, justification and transfer. Single-select items ask for one defensible answer; multiselect items require you to evaluate every statement independently. A union can be a subspace when one component contains the other; an automatic rejection misses that case. Work through the calculation or argument before checking the explanation, then use the feedback to identify the step or assumption that needs attention. This microlearning forms part of Module 5, Vector Spaces and the Four Fundamental Subspaces, in the Complete Linear Algebra Course. Use it to connect intersections and unions with the surrounding chapter and to revisit the reasoning after an initial attempt.

linear algebra

FREE

Start Quiz

5.2-05 Sums of subspaces

Interpret the sum of two subspaces as all combinations of one vector from each. Work with spans and coordinate equations to see how new directions arise and why this construction produces the smallest subspace containing both inputs. What you will practise Build spanning lists for sums. Check membership by solving for components. Use containment and outside directions to identify the resulting space. Preparation and approach Before starting, review spanning sets and the subspace test. The questions progress from recognition and direct applications to error diagnosis, justification and transfer. Single-select items ask for one defensible answer; multiselect items require you to evaluate every statement independently. The sum of subspaces usually contains more vectors than their set-theoretic union. Work through the calculation or argument before checking the explanation, then use the feedback to identify the step or assumption that needs attention. This microlearning forms part of Module 5, Vector Spaces and the Four Fundamental Subspaces, in the Complete Linear Algebra Course. Use it to connect sums of subspaces with the surrounding chapter and to revisit the reasoning after an initial attempt.

linear algebra

FREE

Start Quiz

5.2-06 Direct sums

Explore direct sums through the uniqueness of component decompositions. Examples involving lines, polynomial terms and several summands connect the definition to intersections, proof arguments and the limitations of pairwise tests. What you will practise Solve for components in a two-subspace decomposition. Use intersections to test directness. Diagnose a three-summand example with nonunique decomposition. Preparation and approach Before starting, review subspace sums, independence and solving small linear systems. The questions progress from recognition and direct applications to error diagnosis, justification and transfer. Single-select items ask for one defensible answer; multiselect items require you to evaluate every statement independently. Directness does not require orthogonality, and pairwise zero intersections are insufficient for several summands. Work through the calculation or argument before checking the explanation, then use the feedback to identify the step or assumption that needs attention. This microlearning forms part of Module 5, Vector Spaces and the Four Fundamental Subspaces, in the Complete Linear Algebra Course. Use it to connect direct sums with the surrounding chapter and to revisit the reasoning after an initial attempt.

linear algebra

FREE

Start Quiz

5.2-07 Dimensions of sums

Use the dimension formula for sums and intersections as both a calculation tool and a structural check. The practice includes recovering missing dimensions, enforcing ambient bounds and explaining why shared basis directions are counted only once. What you will practise Calculate dimensions with overlap. Identify impossible subspace specifications. Follow a basis-extension proof of the sum formula. Preparation and approach Before starting, review subspace sums, intersections and finite bases. The questions progress from recognition and direct applications to error diagnosis, justification and transfer. Single-select items ask for one defensible answer; multiselect items require you to evaluate every statement independently. Adding dimensions without accounting for shared directions can exceed the ambient dimension. Work through the calculation or argument before checking the explanation, then use the feedback to identify the step or assumption that needs attention. This microlearning forms part of Module 5, Vector Spaces and the Four Fundamental Subspaces, in the Complete Linear Algebra Course. Use it to connect dimensions of sums with the surrounding chapter and to revisit the reasoning after an initial attempt.

linear algebra

FREE

Start Quiz