To prove that $-(-v) = v$ for every $v$ in a vector space $V$, we will use the properties of vector spaces, particularly the property of additive inverses.

Step 1: Understand the definition of the additive inverse

In a vector space $V$, for any vector $v$, there exists a vector $-v$ such that: $v + (-v) = 0$ where $0$ is the zero vector in $V$.

Step 2: Apply the definition of the additive inverse to $-v$

Now consider the vector $-v$. By the definition of the additive inverse, there exists a vector $-(-v)$ such that: $(-v) + (-(-v)) = 0$

Step 3: Add $v$ to both sides of the equation

To isolate $-(-v)$, we add $v$ to both sides of the equation: $v + \left[(-v) + (-(-v))\right] = v + 0$

Step 4: Use the associative property of vector addition

The associative property of vector addition states that: $(v + (-v)) + (-(-v)) = v$ Since $v + (-v) = 0$, the equation simplifies to: $0 + (-(-v)) = v$

Step 5: Simplify the equation

Adding the zero vector to any vector $w$ in $V$ results in the vector $w$: $-(-v) = v$

Conclusion

We have shown that $-(-v) = v$ for every $v$ in the vector space $V$, as required.

Would you like further details or have any questions? Here are some related problems to consider:

1. Prove that the additive inverse of the zero vector is the zero vector itself.
2. Show that the additive inverse in a vector space is unique.
3. Prove that for any vector $v$ in a vector space $V$, the equation $v + v = 2v$ holds.
4. Demonstrate that the scalar multiplication $0 \cdot v = 0$ for any vector $v$ in a vector