To prove this, we need to carefully consider the setup of the problem.

Given:

• $a \in F$, where $F$ is a field.
• $v \in V$, where $V$ is a vector space over the field $F$.
• The equation $av = 0$, where $0$ represents the zero vector in $V$.

What we want to prove:

We need to show that either:

• $a = 0$ (the scalar is zero), or
• $v = 0$ (the vector is zero).

Step-by-Step Proof:

1. Case 1: If $a = 0$
   If $a = 0$, then the equation becomes: $0 \cdot v = 0$ This is true for any vector $v \in V$, since multiplying the zero scalar with any vector results in the zero vector. Therefore, in this case, the equation $av = 0$ holds trivially for any $v \in V$.

2. Case 2: If $v = 0$
   If $v = 0$, then the equation becomes: $a \cdot 0 = 0$ This is also trivially true for any scalar $a \in F$, since any scalar multiplied by the zero vector results in the zero vector.

3. Case 3: $a \neq 0$
   If $a \neq 0$, we assume the scalar is non-zero. Since $a$ is in the field $F$, and fields have the property that non-zero elements are invertible, there exists an inverse element $a^{-1} \in F$ such that: $a^{-1} a = 1$ Now, multiply both sides of the equation $av = 0$ by $a^{-1}$: $a^{-1} (a v) = a^{-1} \cdot 0$ On the left-hand side, by associativity of scalar multiplication, we get: $(a^{-1} a) v = 0$ Since $a^{-1} a = 1$, this simplifies to: $1 \cdot v = 0$ Therefore, $v = 0$.

Conclusion:

We have shown that if $a \neq 0$, then $v = 0$. Thus, the equation $av = 0$ holds if and only if either:

• $a = 0$, or
• $v = 0$.

This completes the proof.

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Would you like further details on any part of this proof? Here are some related questions to deepen your understanding:

1. Can you explain the properties of fields that were crucial for this proof?
2. What happens if the field $F$ is replaced by a ring?
3. How does this proof change if $V$ is not a vector space but a module?
4. Can this proof be extended to cases where $v$ belongs to an infinite-dimensional vector space?
5. What does this result imply about the relationship between scalar and vector multiplication?

Tip: When working with fields, remember that every non-zero element has a multiplicative inverse—a key property that helps in proofs like this one.