Question

We often write vectors in \(\mathbb{R}^{n}\) as rows. Give a spanning set for the zero subspace \(\\{\mathbf{0}\\}\) of \(\mathbb{R}^{n}\)

Short answer

The spanning set for the zero subspace is the empty set, \(\{\}\).

Step-by-step solution

Understanding the Zero Subspace

The zero subspace of a vector space, denoted \(\{\mathbf{0}\}\), consists only of the zero vector. For the space \(\mathbb{R}^n\), the zero vector is \((0, 0, ..., 0)\), where each component of the vector is zero.

Identifying a Spanning Set

A spanning set for a subspace is any set of vectors that can be linearly combined to produce any vector in the subspace. Since the zero subspace contains only the zero vector, we need a set that can produce only the zero vector.

Choosing the Spanning Set

The simplest choice for a spanning set for the zero subspace \(\{\mathbf{0}\}\) is the empty set, \(\{\}\). An empty set, by definition, only spans the zero vector, and no other vectors can be generated from it.

Key concepts

Zero Vector

In the world of vectors, the zero vector plays a unique and crucial role. This vector, denoted as \(\mathbf{0}\) in mathematical texts, is the vector consisting entirely of zeros. In a space such as \(\mathbb{R}^n\), the zero vector is represented as \((0, 0, ..., 0)\), where the number of zeros matches the dimension of the space you’re dealing with.
The zero vector is special because it acts as the additive identity in vector spaces. What does this mean? It means that adding the zero vector to any other vector will not change that other vector. It’s the equivalent of adding zero in standard arithmetic.

• The zero vector is essential in defining vector spaces' structure.
• It serves as the anchor or the starting point of all vector spaces.

In operations like addition and scalar multiplication, the zero vector maintains neutrality, leaving other elements unchanged by its presence.

Spanning Set

A spanning set for a vector space or subspace is a collection of vectors that, through linear combination, can produce any vector within that space. This concept is powerful because it means we can represent a potentially infinite set of vectors with a finite and specific set of vectors.
To determine what a spanning set is for any space, you need to understand the vectors within it. In the context of the zero subspace \( \{\mathbf{0}\} \), the challenge is unique because this subspace contains only the zero vector.
The minimal spanning set for the zero subspace is, paradoxically, the empty set \( \{\} \). It might seem counterintuitive, but since we only need to generate the zero vector and nothing else, an empty set suffices, ensuring no other vectors are part of the space.

• Spanning sets can be of different sizes, often chosen based on simplicity or computational ease.
• The empty set serves as a spanning set only for the zero subspace, illustrating the unique nature of such spaces.

Linear Combination

The concept of a linear combination is a cornerstone of linear algebra. It involves creating a new vector by adding scaled versions of other vectors. This is expressed as \( c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \cdots + c_k\mathbf{v}_k \), where \(c_i\) are scalars and \(\mathbf{v}_i\) are vectors.
To achieve a linear combination of the zero vector, we seek an expression where the sum results in the zero vector. For example, using the zero vector itself, any scalar multiplied by it will still give the zero vector: \(c\mathbf{0} = \mathbf{0}\). This property highlights how unique the zero vector is in its behavior.
In defining a subspace like the zero subspace, linear combinations become crucial because if you pick vectors from the spanning set (in this case, the empty set), and combine them, you must still end up back at something that qualifies as part of the subspace.

• Linear combinations allow the transformation of a given set of vectors into diverse outputs.
• In the zero subspace's case, they always simplify back to zero, hence the role of the empty set as its spanning set.

Vector Space

A vector space is a collection of vectors that can be added together and multiplied by scalars, producing another vector within the same space. The rules that define vector spaces ensure consistency and linearity in operations.
Vector spaces can take various forms, from simple lines to complex multi-dimensional spaces like \(\mathbb{R}^n\). Each vector space contains its own zero vector, which behaves uniquely by being the only vector that can be reduced to zero through linear combination of itself.
A zero subspace is a subset of a vector space containing only the zero vector. It’s an essential structure that helps in understanding larger and more complex vector spaces by offering a baseline that exemplifies simplicity.

• All vector spaces must contain the zero vector for them to qualify as a vector space.
• Operations within vector spaces (addition, scalar multiplication) always yield results within the space, showing consistency.

Understanding vector spaces and their subspaces challenges students to think abstractly, yet foundationally.