Alphabeta Math

Vector Spaces and Linear Subspaces: Examples and Counterexamples

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✓ 8 results · all verified · 6 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 2 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

ℝ is a vector space over itself, over the embedded copy of ℚ by restriction of scalars, and over ℚ itself via the embedding

Example

In F³ the three coordinate lines are linear subspaces whose internal direct sum is F³, and F⁰ is the zero space

Example

F^ℕ is a vector space and the eventually zero families form a linear subspace of it that is the span of the standard unit families

Counterexample

The first quadrant of ℝ² contains 0 and is closed under addition and is not a linear subspace, since it is not closed under multiplication by -1

Counterexample

The union of the two coordinate axes of F² is closed under scalar multiplication and is not closed under addition, so neither closure condition implies the other

Counterexample

Three lines in F² that meet pairwise only in 0 and whose sum is F² with decompositions that are not unique, so pairwise trivial intersection does not give a direct sum

Example

Two planes in F³ whose sum is F³ and whose intersection is a line, computed explicitly

False statement

FALSE: The union of two linear subspaces is a linear subspace

Sources