Alphabeta Math

Bases and Dimension: Examples and Counterexamples

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✓ 8 results · all verified · 7 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 1 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
Linear Algebra
Real Analysis
Set Theory
2 · Logical flowchart

3 · Items

Example

The vector (1,2) ∈ ℝ² has coordinate list (1,2) in the standard ordered basis, (2,1) in its reversal, and (2,-1) in the ordered basis ((1,1),(1,0))

Example

The standard unit families eₖ ∈ F^ℕ form a basis of the linear subspace of eventually zero families: an explicit infinite basis, built with no choice principle

Example

ℝ as a vector space over ℚ has a basis, and every such basis is infinite; the existence proof exhibits none

Counterexample

The standard unit families { eᵢ : i ∈ ℕ } are linearly independent in F^ℕ but do not span it: the constant family 1_F is not a finite linear combination of them

Counterexample

{(1,0), (0,1), (1,1)} spans F² and is linearly dependent, so a spanning set need not be a basis; each of its three two-element subsets is a basis

Counterexample

Inside the space of eventually zero families, the linear subspace spanned by { eᵢ : i ≥ 1 } is proper and has a basis equinumerous with a basis of the whole space, so "equal dimension forces equality" fails without finite dimension

Counterexample

Three distinct lines U₀, U₁, U₂ in F² have dim_F(U₀+U₁+U₂) = 2 while the inclusion-exclusion analogue of the dimension formula predicts 3, so the two-subspace formula does not extend

False statement

FALSE: the union of two linearly independent subsets of a vector space is linearly independent

Sources