Yes, the collection $a + w, b + x, c + y, d + z$ spans $V$. Here’s why:

Given:

• The vectors $a, b, c, d$ span $V$.
• The vectors $w, x, y, z$ span $V$. This means:
• Any vector in $V$ can be written as a linear combination of $a, b, c, d$.
• Any vector in $V$ can also be written as a linear combination of $w, x, y, z$.

Objective:

Prove that the set $\{a + w, b + x, c + y, d + z\}$ spans $V$.

Proof Outline:

Since $a, b, c, d$ span $V$, any vector $v \in V$ can be written as: $v = \alpha_1 a + \alpha_2 b + \alpha_3 c + \alpha_4 d$ for some scalars $\alpha_1, \alpha_2, \alpha_3, \alpha_4 \in \mathbb{R}$.

Similarly, since $w, x, y, z$ also span $V$, any vector $v' \in V$ can be written as: $v' = \beta_1 w + \beta_2 x + \beta_3 y + \beta_4 z$ for some scalars $\beta_1, \beta_2, \beta_3, \beta_4 \in \mathbb{R}$.

Step-by-Step Argument:

Consider any vector $v \in V$. Because both sets $\{a, b, c, d\}$ and $\{w, x, y, z\}$ span $V$, we know that: $v = \alpha_1 a + \alpha_2 b + \alpha_3 c + \alpha_4 d$ and $v = \beta_1 w + \beta_2 x + \beta_3 y + \beta_4 z$ Now, observe that you can combine the vectors in the set $\{a+w, b+x, c+y, d+z\}$ linearly as follows: $(\alpha_1)(a + w) + (\alpha_2)(b + x) + (\alpha_3)(c + y) + (\alpha_4)(d + z)$ This expands as: $\alpha_1 a + \alpha_1 w + \alpha_2 b + \alpha_2 x + \alpha_3 c + \alpha_3 y + \alpha_4 d + \alpha_4 z$ Which can be written as: $(\alpha_1 a + \alpha_2 b + \alpha_3 c + \alpha_4 d) + (\alpha_1 w + \alpha_2 x + \alpha_3 y + \alpha_4 z)$ Since $a, b, c, d$ span $V$, and $w, x, y, z$ span $V$, this is simply a linear combination of the vectors from both sets. Therefore, any vector in $V$ can be written as a linear combination of the vectors $a + w, b + x, c + y, d + z$.

Thus, the set $\{a + w, b + x, c + y, d + z\}$ spans $V$.

Conclusion:

The collection $a + w, b + x, c + y, d + z$ indeed spans $V$.

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Let me know if you'd like further clarification or additional details!

Follow-up Questions:

1. How can we prove that a smaller subset of vectors spans a vector space?
2. What happens if we have fewer than 4 vectors but still span $V$?
3. Can we say the same about the span of vectors in complex vector spaces?
4. How do you prove that two different sets of vectors span the same space?
5. If a set of vectors does not span a space, what can we say about their linear dependence?

Tip:

Whenever you're dealing with spanning sets, remember that a linear combination of vectors from two different spanning sets will always span the same space.