Theorem (Basic Properties of Vector Spaces): Let $V$ be a vector space over $F$. Then:
$0v = 0$ — The scalar zero times any vector equals the zero vector.
$a0 = 0$ — Any scalar times the zero vector equals the zero vector.
$(-1)v = -v$ — Multiplying by $-1$ gives the additive inverse.
If $av = 0$, then $a = 0$ or $v = 0$ — No zero divisors.
The zero vector is unique.
Additive inverses are unique.
Proof sketch for (1): $0v = (0+0)v = 0v + 0v$ by D2. Adding $-(0v)$ to both sides gives $0 = 0v$. $\blacksquare$
Proof sketch for (4): If $a \neq 0$, multiply both sides of $av = 0$ by $a^{-1}$: $a^{-1}(av) = a^{-1} \cdot 0$, so $(a^{-1}a)v = 0$, hence $1 \cdot v = 0$, so $v = 0$. $\blacksquare$
Application: These properties are used constantly in proofs. For instance, property (4) is key in proving that eigenvectors corresponding to distinct eigenvalues are linearly independent.
**Theorem (Basic Properties of Vector Spaces):** Let $V$ be a vector space over $F$. Then:
1. **$0v = 0$** — The scalar zero times any vector equals the zero vector.
2. **$a0 = 0$** — Any scalar times the zero vector equals the zero vector.
3. **$(-1)v = -v$** — Multiplying by $-1$ gives the additive inverse.
4. **If $av = 0$, then $a = 0$ or $v = 0$** — No zero divisors.
5. **The zero vector is unique.**
6. **Additive inverses are unique.**
**Proof sketch for (1):** $0v = (0+0)v = 0v + 0v$ by D2. Adding $-(0v)$ to both sides gives $0 = 0v$. $\blacksquare$
**Proof sketch for (4):** If $a \neq 0$, multiply both sides of $av = 0$ by $a^{-1}$: $a^{-1}(av) = a^{-1} \cdot 0$, so $(a^{-1}a)v = 0$, hence $1 \cdot v = 0$, so $v = 0$. $\blacksquare$
**Application:** These properties are used constantly in proofs. For instance, property (4) is key in proving that eigenvectors corresponding to distinct eigenvalues are linearly independent.