Suppose V is a vector space over a eld F. Show that if v ∈ V and c ∈ F, then (−c)v = −(cv)

To prove the statement \((−c)v = −(cv)\) for a vector \(v \in V\) in a vector space \(V\) over a field \(F\) and a scalar \(c \in F\), we can proceed as follows:

### Proof:

1. **Recall the properties of scalar multiplication:**
   - Distributive property: \( a(v_1 + v_2) = av_1 + av_2 \) for \(a \in F\) and \(v_1, v_2 \in V\).
   - Distributive property over field addition: \( (a + b)v = av + bv \) for \(a, b \in F\) and \(v \in V\).
   - Scalar multiplication associativity: \( (ab)v = a(bv) \) for \(a, b \in F\) and \(v \in V\).
   - Scalar multiplication identity: \( 1v = v \) for \(v \in V\) and \(1 \in F\) (where \(1\) is the multiplicative identity in \(F\)).
   - Scalar multiplication with zero: \( 0v = 0_V \) (where \(0_V\) is the zero vector in \(V\)).

2. **Consider the expression \((−c)v\):**

   We know that \( −c \) is the additive inverse of \( c \) in \(F\). By definition of scalar multiplication and the properties of the field, we have:
   \[
   (−c)v + (cv) = (−c + c)v = 0v = 0_V
   \]
   This equation tells us that \((−c)v\) is the additive inverse of \(cv\). In other words:
   \[
   (−c)v = −(cv)
   \]
   where \(−(cv)\) denotes the additive inverse of the vector \(cv\).

### Conclusion:
Thus, we have shown that \((−c)v = −(cv)\) for any vector \(v \in V\) and scalar \(c \in F\).

Would you like more details or have any questions?

Here are 5 related questions you might consider:
1. How does the proof change if \(V\) is over a ring instead of a field?
2. Can you show that \(0 \cdot v = 0_V\) using the properties of a vector space?
3. What are the implications of the scalar multiplication properties on the structure of the vector space?
4. How does the proof apply if \(v\) is the zero vector?
5. What role does the distributive property play in the proof?

**Tip:** When proving identities in vector spaces, always rely on the axioms and properties of vector spaces, such as distributive properties and associativity, to derive the required result.

Explore a detailed proof showing how scalar multiplication behaves in vector spaces. Understand the relationship between scalar multiplication and vector addition, using field properties. Suitable for advanced undergraduate students and beyond.

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