Prove or give a counterexample: If 𝑣1
, 𝑣2
, 𝑣3
, 𝑣4
spans 𝑉, then the list
𝑣1 − 𝑣2
, 𝑣2 − 𝑣3
, 𝑣3 − 𝑣4
, 𝑣4
also spans �

### Problem Statement:
We are asked to determine whether the list:
\[
v_1 - v_2, \quad v_2 - v_3, \quad v_3 - v_4, \quad v_4
\]
spans the vector space \( V \), given that the original set of vectors \( v_1, v_2, v_3, v_4 \) spans \( V \).

### Solution Strategy:
- **Definition of Spanning Set**: A set of vectors \( \{u_1, u_2, \dots, u_k\} \) spans a vector space \( V \) if every vector in \( V \) can be written as a linear combination of these vectors.
- The goal is to check whether the list \( v_1 - v_2, v_2 - v_3, v_3 - v_4, v_4 \) forms a spanning set for \( V \), under the assumption that \( v_1, v_2, v_3, v_4 \) spans \( V \).

### Step 1: Express the Original Vectors
We will express the original set \( \{v_1, v_2, v_3, v_4\} \) as linear combinations of the new set \( \{v_1 - v_2, v_2 - v_3, v_3 - v_4, v_4\} \) to check if the new set spans \( V \).

- First, we express each of the original vectors \( v_1, v_2, v_3, v_4 \) in terms of the new vectors:
  \[
  v_4 = v_4
  \]
  \[
  v_3 = (v_3 - v_4) + v_4
  \]
  \[
  v_2 = (v_2 - v_3) + (v_3 - v_4) + v_4
  \]
  \[
  v_1 = (v_1 - v_2) + (v_2 - v_3) + (v_3 - v_4) + v_4
  \]

### Step 2: Check if the New Set Spans \( V \)
Now we need to verify if the vectors \( v_1 - v_2, v_2 - v_3, v_3 - v_4, v_4 \) span \( V \). Since \( v_1, v_2, v_3, v_4 \) span \( V \), any vector in \( V \) can be written as a linear combination of these vectors. If we can express \( v_1, v_2, v_3, v_4 \) in terms of the new set, then the new set also spans \( V \).

From the expressions above, it is clear that each \( v_i \) can be written as a linear combination of \( v_1 - v_2, v_2 - v_3, v_3 - v_4, v_4 \). Therefore, the new set \( \{v_1 - v_2, v_2 - v_3, v_3 - v_4, v_4\} \) spans \( V \).

### Conclusion:
The list \( v_1 - v_2, v_2 - v_3, v_3 - v_4, v_4 \) **also spans** the vector space \( V \). Thus, the statement is **true**.

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Would you like a more detailed breakdown or have any questions?

Here are 5 related questions to deepen your understanding:
1. What conditions are necessary for a set of vectors to form a spanning set?
2. Can you find another way to represent the vectors \( v_1, v_2, v_3, v_4 \) in terms of the new set?
3. How can we determine whether a set of vectors is linearly independent?
4. What is the geometric interpretation of a spanning set in \( \mathbb{R}^3 \) or \( \mathbb{R}^4 \)?
5. If the original set \( v_1, v_2, v_3, v_4 \) were not linearly independent, what would change in this proof?

**Tip**: When checking if a set spans a space, always try to express the original basis in terms of the new vectors. If you can, then the new set also spans the space.

Prove or give a counterexample: If v1, v2, v3, v4 spans V, then the list v1 − v2, v2 − v3, v3 − v4, v4 also spans V.

Determine whether the modified vector list v1 - v2, v2 - v3, v3 - v4, v4 spans the vector space V, given that v1, v2, v3, v4 spans V. This problem explores key concepts in linear algebra, such as spanning sets and linear combinations. Suitable for undergraduate students studying Linear Algebra.

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