To solve the problem using the rules and theorems from Schaum’s Outline of Linear Algebra by Lipschutz and Lipson, I'll go through each part carefully, justifying the solution step by step.

Problem Statement:

We are given a vector space $V$, and we need to prove two properties:

1. For every $k \in \mathbb{R}$ and every $u, v \in V$, we have: $k(u - v) = ku - kv$
2. For every vector $v \in V$, we have: $v + v = 2v$

Part 1: $k(u - v) = ku - kv$

This is a distributive property of scalar multiplication over vector subtraction, and we can justify each step using the axioms of vector spaces as outlined in Schaum’s Outline:

Step-by-Step Justification:

1. Vector subtraction in terms of addition:
   Recall that in a vector space, subtraction is defined as adding the additive inverse: $u - v = u + (-v)$ where $-v$ is the additive inverse of $v$ (i.e., $v + (-v) = 0$).

2. Distribute scalar $k$ over the sum:
   By the distributive property of scalar multiplication over vector addition (vector space axiom): $k(u + (-v)) = ku + k(-v)$

3. Simplify using scalar multiplication and inverses:
   By the property of scalar multiplication, we know that $k(-v) = -(kv)$ (the scalar multiplied by the additive inverse is the additive inverse of the scalar product): $ku + k(-v) = ku - kv$

Thus, we have shown that $k(u - v) = ku - kv$, as required.

Part 2: $v + v = 2v$

This property is based on the definition of scalar multiplication.

Step-by-Step Justification:

1. Express $2v$ as scalar multiplication:
   By definition, the scalar $2$ applied to a vector $v$ is: $2v = v + v$

2. Interpretation in terms of addition:
   In a vector space, multiplying a vector $v$ by a scalar $2$ means adding $v$ to itself: $v + v = 2v$ This is derived directly from the definition of scalar multiplication in the vector space axioms.

Hence, we have shown that $v + v = 2v$, as required.

Conclusion:

Using vector space properties from Schaum's Outline, we've carefully proven both parts:

1. $k(u - v) = ku - kv$ by using the distributive property of scalar multiplication over addition and the property of scalar multiplication with inverses.
2. $v + v = 2v$ by the definition of scalar multiplication as repeated addition.

Would you like to dive deeper into any of these concepts or have more detailed explanations?

Here are 5 related questions you might find useful to expand on these ideas:

1. How can we use the distributive property for more complex scalar and vector expressions?
2. What are the axioms of a vector space, and how do they ensure properties like these?
3. How would these proofs change if we were working in a field other than $\mathbb{R}$, such as $\mathbb{C}$?
4. How does the concept of linearity relate to the distributive properties of scalar multiplication?
5. Can you generalize these proofs for matrices and matrix multiplication?

Tip: Understanding the axioms of a vector space is crucial as they form the foundation for all properties and operations within linear algebra.