A VECTOR SPACE $V$ over a field $\mathbb{F}$ requires which axiom about scalar multiplication?
A$\alpha(\beta v) = (\alpha\beta)v$
B$\alpha v = v$ for every $\alpha$
CScalars must be integers
DVectors must be linearly independent
Answer & Solution
Correct answer: A. $\alpha(\beta v) = (\alpha\beta)v$
1. A vector space has axioms for addition and scalar multiplication.
2. Scalar mult axiom: $\alpha(\beta v) = (\alpha\beta) v$ — associativity of scalar action.
3. Other scalar mult axioms: $1 \cdot v = v$, $\alpha(u + v) = \alpha u + \alpha v$, $(\alpha + \beta) v = \alpha v + \beta v$.
4. Option B has the WRONG identity (would force every vector to equal itself for ANY scalar). Options C, D are unrelated.
_Source: Sergei Treil, "Linear Algebra Done Wrong", §1.1 (Vector spaces — axioms)._
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