On \(\RR^n\), define two operations \[ \alpha \oplus \beta = \alpha - \beta, \qquad c \cdot \alpha = -c\alpha. \] The operations on the right are the usual ones. Which of the axioms for a vector space are satisfied by \((\RR^n, \oplus, \cdot)\)?
Solution
The operation \(\oplus\) is not a vector addition. For instance, \[ \alpha\oplus\beta=\alpha-\beta, \qquad \beta\oplus\alpha=\beta-\alpha, \] which are generally unequal, so commutativity fails. Associativity also fails: \[ (\alpha\oplus\beta)\oplus\gamma=\alpha-\beta-\gamma, \] while \[ \alpha\oplus(\beta\oplus\gamma)=\alpha-\beta+\gamma. \] There is a right identity \(0\) because \(\alpha\oplus0=\alpha\), but no two-sided additive identity since \(0\oplus\alpha=-\alpha\) in general.
For scalar multiplication, one has \[ 1\cdot\alpha=-\alpha, \] so the axiom \(1\cdot\alpha=\alpha\) fails. Also \[ (ab)\cdot\alpha=-ab\alpha, \qquad a\cdot(b\cdot\alpha)=ab\alpha, \] so scalar associativity fails in general.
Some distributive identities do hold accidentally, but the vector-space axioms as a whole fail; in particular \((\RR^n,\oplus,\cdot)\) is not a vector space.