Let's break down and analyze the two statements.

Statement 1:

"If $a, b, c, d$ spans $V$ and $w, x, y, z$ spans $V$, then the collection $a, b, c, d, w, x, y, z$ spans $V$."

Analysis:

• If $a, b, c, d$ spans $V$, it means that any vector in $V$ can be expressed as a linear combination of $a, b, c,$ and $d$.
• Similarly, if $w, x, y, z$ spans $V$, it means any vector in $V$ can also be expressed as a linear combination of $w, x, y,$ and $z$.

When you combine the two sets $\{a, b, c, d\}$ and $\{w, x, y, z\}$ into one set $\{a, b, c, d, w, x, y, z\}$, the combined set will still span $V$ as long as the vectors in the set are linearly independent or at least one of the two sets already spans the whole space.

Since both $\{a, b, c, d\}$ and $\{w, x, y, z\}$ independently span $V$, their union will also span $V$ because it still includes enough vectors to express any vector in $V$.

Conclusion:

The first statement is true.

Statement 2:

"If $a, b, c, d$ spans $V$ and $w, x, y, z$ spans $V$, then the collection $a, b, c, d, w$ spans $V$."

Analysis:

• The set $a, b, c, d$ spans $V$, so any vector in $V$ can be written as a linear combination of $a, b, c, d$.
• The set $w, x, y, z$ also spans $V$, meaning any vector in $V$ can also be written as a linear combination of $w, x, y, z$.

However, in this case, you are only taking the first vector from the second set, namely $w$, and combining it with the set $\{a, b, c, d\}$.

To determine whether the collection $\{a, b, c, d, w\}$ spans $V$, we need to consider the possibility that the set $\{w\}$ does not add anything new to the span of $\{a, b, c, d\}$. For example, if $w$ can already be expressed as a linear combination of $a, b, c, d$, then adding $w$ doesn't increase the span of the set.

Thus, while $a, b, c, d$ spans $V$, the set $\{a, b, c, d, w\}$ might not span $V$ unless $w$ is linearly independent of the other vectors in $\{a, b, c, d\}$.

Conclusion:

The second statement is not necessarily true. It depends on whether $w$ is linearly independent of $a, b, c, d$. If $w$ can be written as a linear combination of the other vectors, then the collection $\{a, b, c, d, w\}$ will not span $V$.

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To summarize:

1. The first statement is true.
2. The second statement is false unless $w$ is linearly independent of $a, b, c, d$.

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Would you like a more detailed explanation or an example to clarify this further? Here's a set of related questions for deeper understanding:

1. How do you check if a set of vectors spans a vector space?
2. What is the concept of linear independence, and how does it relate to spanning a space?
3. Can a single vector span a vector space? If so, under what conditions?
4. How does the dimension of a vector space affect the number of vectors required to span it?
5. What happens if you add a redundant vector to a spanning set?

Tip: Always check for linear dependence or independence when combining sets of vectors. If the new vector is already in the span of the previous ones, it doesn't increase the span.