5.3 Independence, Basis and Dimension

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

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Oct 20, 2026

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5.3-01 Independence in abstract spaces

Prove independence in polynomial, matrix and function spaces using coefficients, evaluations and explicit relations. The exercises emphasize that function identities depend on the domain and that sampled values can certify independence only under the right conditions. What you will practise Compare polynomial coefficients and matrix entries. Use strategically chosen function evaluations. Distinguish a global identity from agreement at a few inputs. Preparation and approach Before starting, review linear combinations, polynomial algebra and elementary functions. 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. Different formulas or nonzero vectors need not be independent; the entire zero relation must be checked. 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 independence in abstract spaces with the surrounding chapter and to revisit the reasoning after an initial attempt.

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5.3-02 Verifying a basis

Verify a basis by checking both sufficient coverage and absence of redundancy. Polynomials, matrices, planes and function spans provide varied settings for calculations, counterexamples and parameter-dependent basis decisions. What you will practise Separate spanning from independence. Check candidates belong to the stated space. Solve a parameter case and justify a function-space basis. Preparation and approach Before starting, review span, independence and elementary 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. Matching the expected number of candidates is not enough unless membership and a basis condition are established. 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 verifying a basis with the surrounding chapter and to revisit the reasoning after an initial attempt.

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5.3-03 Unique representation

Connect the two basis conditions to existence and uniqueness of representations. This short set combines coefficient calculations with subtraction proofs and a redundant spanning list whose representations reveal a free parameter. What you will practise Identify which property supplies existence. Prove uniqueness by subtracting expansions. Track changes caused by reordering a basis. Preparation and approach Before starting, review spanning, independence and two-variable equations. 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 spanning list can supply many representations of the same vector when it contains redundancy. 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 unique representation with the surrounding chapter and to revisit the reasoning after an initial attempt.

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5.3-04 The exchange principle

Use the exchange principle to compare independent and spanning lists and to replace basis vectors safely. The practice develops the nonzero-coefficient argument behind replacement, equal basis sizes, extraction and extension. What you will practise Choose valid basis replacements. Compare the sizes of independent and spanning lists. Recover original basis vectors after a replacement. Preparation and approach Before starting, review basis expansions and solving linear relations. 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 replacement is justified by its coefficient in a basis expansion, not simply by the new vector being nonzero. 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 exchange principle with the surrounding chapter and to revisit the reasoning after an initial attempt.

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5.3-05 Dimension

Calculate dimension across coordinate, polynomial and matrix spaces while keeping track of the scalar field. The set also examines constraints, proper subspaces and examples whose finite parameter counts differ from the apparent size of their objects. What you will practise Count independent coefficients or entries. Use homogeneous constraints to find free parameters. Compare finite-dimensional spaces and unbounded polynomial spaces. Preparation and approach Before starting, review bases, polynomial degree and matrix symmetry. 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. Coordinate length, polynomial degree and the number of elements in a space are different quantities. 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 dimension with the surrounding chapter and to revisit the reasoning after an initial attempt.

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