Question

Describe span \(\\{\mathbf{0}\\}\).

Short answer

The span of \(\{\mathbf{0}\}\) is \(\{\mathbf{0}\}\).

Step-by-step solution

Understanding the Concept of Span

The span of a set of vectors is the collection of all possible linear combinations of those vectors. A linear combination of vectors involves adding together scalar multiples of those vectors.

Identifying the Given Set of Vectors

In this case, we are given a single vector set containing the zero vector \(\{\mathbf{0}\}\). The zero vector is a vector where every component is zero, denoted by \(\mathbf{0} = (0, 0, ..., 0)\).

Analyzing Linear Combinations

For the set \(\{\mathbf{0}\}\), any linear combination will look like \(c \cdot \mathbf{0}\), where \(c\) is a scalar. Since multiplying any scalar by zero still gives zero, any linear combination of \(\{\mathbf{0}\}\) results in \(\mathbf{0}\).

Concluding the Span

Since the only vector that can be constructed as a linear combination of \(\mathbf{0}\) is \(\mathbf{0}\) itself, the span of the set \(\{\mathbf{0}\}\) is simply the zero vector itself. Hence, \(\text{span}\{\mathbf{0}\} = \{\mathbf{0}\}\).

Key concepts

Vector Space

In linear algebra, a vector space is a fundamental concept that combines geometry and algebra. It refers to a set of vectors where two main operations are defined: vector addition and scalar multiplication. These operations must satisfy a number of axioms or rules, such as associativity, commutativity of addition, the existence of an additive identity (often called the zero vector), and the existence of additive inverses.

To visualize a vector space, imagine it as an entire "world" or "environment" where all vectors in that space can exist. This space is stable under both vector addition and scalar multiplication. Therefore, any linear combination of vectors within that world still stays within the same "world".

• The set of vectors forms an abelian group under addition.
• Scalar multiplication is distributive across both vector addition and field addition.
• There exists a multiplicative identity over scalar multiplication.

An example of a vector space is \( ext{R}^n\), which is the collection of all n-tuples of real numbers. Whether dealing with two-dimensional, three-dimensional, or higher-dimensional spaces, vector spaces are crucial in understanding transformations and systems of linear equations.

Zero Vector

The zero vector is a unique and special type of vector in a vector space. Symbolized by \(\mathbf{0}\), it has all of its components equal to zero, such as \(\mathbf{0} = (0, 0, ..., 0)\). This vector is the "silent hero" of linear algebra because it acts as the additive identity. This means that adding \(\mathbf{0}\) to any vector \(\mathbf{v}\) in the vector space produces \(\mathbf{v}\) itself: \(\mathbf{v} + \mathbf{0} = \mathbf{v}\).

Understanding the zero vector is critical because:

• It serves as the base case in many linear combinations.
• Every vector space has exactly one zero vector.
• It helps in defining vector subspaces and matrix null spaces.

Even though the zero vector seems simple, it plays a pivotal role in linear combinations, as you'll soon see. In the case of a set containing only the zero vector, like \(\{\mathbf{0}\}\), its span is exceedingly straightforward—it can only generate itself, making its span \(\{\mathbf{0}\}\).

Linear Combination

Linear combinations are statements of great importance in linear algebra that involve creating a new vector from a given set of vectors. Essentially, they involve multiplying each vector by a scalar and then adding up these products. For example, if you have vectors \(\mathbf{a}, \mathbf{b}\), and scalars \(c_1, c_2\), a linear combination would look like \(c_1\mathbf{a} + c_2\mathbf{b}\).

This is what makes linear combinations so powerful:

• Every time you modify the scalars, you achieve different results.
• They allow you to span entire vector spaces, given a sufficient number of independent vectors.
• Linear combinations are foundational in solving systems of linear equations through matrix representations.

When it comes to a set containing the zero vector \(\{\mathbf{0}\}\), every linear combination formed is still the zero vector, because any scalar times zero is zero, \(c \cdot \mathbf{0} = \mathbf{0}\). Therefore, the exploration of such combinations highlights the boundaries and dynamics of the set's span.