Question

Let \(U\) and \(W\) denote subspaces of \(\mathbb{R}^{n}\) and assume that \(U \subseteq W\). If \(\operatorname{dim} W=1\), show that either \(U=\\{\boldsymbol{0}\\}\) or \(U=W\)

Short answer

Either \( U = \{\boldsymbol{0}\} \) or \( U = W \).

Step-by-step solution

Understand the Problem

We need to analyze the relationship between two subspaces \( U \) and \( W \) of \( \mathbb{R}^{n} \) given that \( U \subseteq W \). We know \( \operatorname{dim} W = 1 \), which means \( W \) is spanned by a single non-zero vector.

Analyze the Possible Dimensions of U

Since \( U \subseteq W \) and \( \operatorname{dim} W = 1 \), the dimension of \( U \), denoted \( \operatorname{dim} U \), must be either 0 or 1. This is because the dimension of a subspace cannot exceed the dimension of the space it is contained within.

Consider \( \operatorname{dim} U = 0 \) Case

If \( \operatorname{dim} U = 0 \), then \( U \) contains only the zero vector, i.e., \( U = \{\boldsymbol{0}\} \). This situation is straightforward as \( U \) must be the trivial subspace since no non-zero vectors are needed to span it.

Consider \( \operatorname{dim} U = 1 \) Case

If \( \operatorname{dim} U = 1 \), then \( U \) is also spanned by a single non-zero vector, the same vector that spans \( W \), given \( U \subseteq W \). Thus, \( U \) must coincide with \( W \), i.e., \( U = W \).

Conclusion

We have deduced that for subspaces \( U \) and \( W \), where \( U \subseteq W \) and \( \operatorname{dim} W = 1 \), either \( U = \{\boldsymbol{0}\} \) or \( U = W \). These are the only two possibilities.

Key concepts

Subspaces

A subspace is a special kind of subset of a vector space that is itself a vector space, satisfying the same conditions of vector addition and scalar multiplication.
In the context of our exercise, we have two subspaces, denoted by \( U \) and \( W \), which are subspaces of \( \mathbb{R}^n \). This simply means they reside within the larger space \( \mathbb{R}^n \), much like a small team being part of a bigger organization.

The important condition here is that \( U \subseteq W \), indicating that \( U \) is contained fully within \( W \). This means every vector in \( U \) is also a vector in \( W \). Given that \( W \) is a subspace, it must satisfy the properties of having a zero vector, being closed under addition, and being closed under scalar multiplication. Since \( U \) is also a subspace, it must satisfy these conventions within the confines of \( W \), making it an interesting part to analyze when it comes to dimensions.

Dimension

The concept of dimension in linear algebra refers to the minimum number of vectors needed to span a space. Think of it like the number of axes you need to define position within the space.

In our original problem, we were given that \( \text{dim} \ W = 1 \), meaning \( W \) is one-dimensional. This indicates that \( W \) can be completely described using just a single non-zero vector. Hence, the entire subspace being just a straight line in the context of \( \mathbb{R}^n \). For \( U \), since \( U \subseteq W \), the dimension of \( U \), \( \text{dim} \ U \), can only be 0 or 1. Any attempt to add more dimensions would push \( U \) out of the constraints of \( W \).

A 0-dimensional subspace, like \( U = \{ \boldsymbol{0} \} \), contains only the zero vector. It's the smallest subspace possible. A 1-dimensional \( U \), however, matches \( W \), confirming \( U = W \). The evident conclusion is formed based on these limits.

Vector Spaces

Vector spaces are foundational in linear algebra and provide a framework in which vectors operate.
A vector space is a collection of vectors where two main operations occur: vector addition and scalar multiplication. These operations must satisfy a set of rules called axioms, including presence of a zero vector, associativity, commutativity, and distributive laws.

• Zero Vector: Exists as a neutral addition element.
• Associativity and Commutativity: Allow vectors to be added in a familiar and flexible manner.
• Distributive Laws: Ensure scalar multiplication respects vector addition.

Subspaces, such as \( U \) and \( W \) in our problem, inherit these properties and it is within these confines that they interact.
This exercise showcases how the dimension defines limits on these subspaces within \( \mathbb{R}^n \). Despite their inclusion in a potentially vast space of dimension \( n \), they maintain their functionality and identity, obeying all vector space axioms while interplaying with each other's dimension constraints as seen in the example problem.