In this problem, you are asked to consider three vector subspaces $U$, $V$, and $W$ of $\mathbb{R}^n$ such that $U \subseteq V \subseteq W$. You are given the dimensions of $U$ and $W$ as follows:

• $\dim(U) = 5$
• $\dim(W) = 7$

The task is to determine what can be said about $V$ from this information.

Solution and Analysis

1. Dimension Constraints: Since $U \subseteq V \subseteq W$, the dimension of $V$ must satisfy: $\dim(U) \leq \dim(V) \leq \dim(W)$ Substituting the known dimensions: $5 \leq \dim(V) \leq 7$ Therefore, $V$ could have a dimension of 5, 6, or 7.

2. Evaluating the Statements:

   • "It always contains a set of 7 linearly independent vectors."

     • This is false because $\dim(V)$ could be 5, 6, or 7. If $\dim(V) = 5$ or $\dim(V) = 6$, it cannot contain a set of 7 linearly independent vectors.

   • "It always contains a set of 5 linearly independent vectors."

     • This is true because $\dim(U) = 5$, and since $U \subseteq V$, $V$ must contain at least a set of 5 linearly independent vectors.

   • "The zero vector is not contained in it."

     • This is false because all subspaces contain the zero vector by definition.

   • "It contains a non-zero vector."

     • This is true because $V$ has a dimension of at least 5, which implies it contains more than just the zero vector.

   • "dim V = 6."

     • This is false as $\dim(V)$ could be 5, 6, or 7; it is not necessarily 6.

Conclusion

The correct answers are:

• "It always contains a set of 5 linearly independent vectors."
• "It contains a non-zero vector."

Would you like further details on any part of this solution?

Related Questions

1. What are the properties of subspaces in terms of containment and dimensionality?
2. How do we determine if a vector space contains the zero vector?
3. Why can a subspace of dimension 5 not contain 7 linearly independent vectors?
4. How does the concept of linear independence apply when determining possible vector sets in subspaces?
5. If we know $\dim(V) = 6$, what additional information could be deduced about $V$ in this context?

Tip

When comparing dimensions of subspaces, remember that the dimension of a subspace represents the maximum number of linearly independent vectors it can contain. This helps in setting upper and lower bounds for dimensions in nested subspaces.