Definition of Vector Space

Exercise 1: Double Additive Inverse

Prove that $ -(-v) = v $ for every $ v \in V $, where $ V $ is a vector space.

Exercise 2: Zero Product Property

Suppose $ a \in \mathbb{F} $ (a field), $ v \in V $, and $ av = 0 $. Prove that $ a = 0 $ or $ v = 0 $.

Exercise 3: Unique Solution to Vector Equation

Suppose $ v, w \in V $. Explain why there exists a unique $ x \in V $ such that $ v + 3x = w $.

Exercise 4: Alternative Axiom for Additive Inverses

Show that in the definition of a vector space, the additive inverse condition can be replaced with:

$$ 0v = 0 \text{ for all } v \in V $$

(where left 0 $ \in \mathbb{F}$, right 0 $ \in V $).

Exercise 5: Extended Real Numbers as a Vector Space?

Define addition and scalar multiplication on $ \mathbb{R}^* = \mathbb{R} \cup {\infty, -\infty} $ as follows:

For $ t \in \mathbb{R} $:

$$ t \cdot \infty = \begin{cases} -\infty & \text{if } t < 0, \\ 0 & \text{if } t = 0, \\ \infty & \text{if } t > 0, \end{cases} \quad \quad t \cdot (-\infty) = \text{(analogous)} $$

For $ t \in \mathbb{R} $:

$$ t + \infty = \infty + t = \infty, \quad \infty + \infty = \infty, \quad \infty + (-\infty) = 0. $$

Is $ \mathbb{R}^* $ a vector space over $ \mathbb{R} $? Justify.