Question

We often write vectors in ${\mathbb{R}}^n$ as rows. If $S$ and $T$ are nonempty sets of vectors in ${\mathbb{R}}^n$, and if $S\subseteq T$, show that $$\begin{gathered} \mathrm{span} \\ S \\ \subseteq \mathrm{span} \\ T \end{gathered}$$.

Short answer

Since $S\subseteq T$, $\mathrm{span}(S)\subseteq \mathrm{span}(T)$.

Step-by-step solution

Understand the Problem Statement

We need to show that if one set of vectors $S$ is a subset of another set of vectors $T$ in ${\mathbb{R}}^n$, then the span of $S$, written as $\mathrm{span}(S)$, is also a subset of the span of $T$, written as $\mathrm{span}(T)$.

Define the Vector Span

The span of a set of vectors is the set of all possible linear combinations of those vectors. Therefore, $\mathrm{span}(S)$ is the set containing all vectors that can be formed as linear combinations of vectors in $S$. Similarly, $\mathrm{span}(T)$ is the set of all vectors that can be formed as linear combinations of vectors in $T$.

Use Set Subset Relationship

Since $S\subseteq T$, every vector in $S$ is also in $T$. This means that any linear combination using vectors from $S$ can also be formed using vectors from $T$, as all vectors of $S$ are included in $T$.

Constructing Linear Combinations

Take any vector $\mathbf{v}$ in $\mathrm{span}(S)$. By definition, $\mathbf{v}$ can be expressed as $a_1{\mathbf{s}}_1+a_2{\mathbf{s}}_2+\dots +a_k{\mathbf{s}}_k$ where each ${\mathbf{s}}_i\in S$ and $a_i$ are scalars. Since each ${\mathbf{s}}_i$ is also in $T$ (due to $S\subseteq T$), this linear combination is valid in $\mathrm{span}(T)$.

Conclude the Argument

Since any vector $\mathbf{v}$ that can be formed using vectors from $S$ can also be formed using vectors from $T$, it follows that $\mathrm{span}(S)\subseteq \mathrm{span}(T)$. Thus, we have shown that the span of $S$ is a subset of the span of $T$.

Key concepts

Vector Spaces

In the realm of linear algebra, a vector space is a collection of objects called vectors. These vectors can be added together and multiplied by scalars, which are often real numbers. Vector spaces play a fundamental role in the world of mathematics and physics because they provide the setting for the principles and operations of linear combinations.

Think of a vector space as a vast "arena" where vectors can freely roam, and any linear operation within this space remains within its boundaries. For example, when working in extit{n}-dimensional real space, denoted as extbf{R} ^{n}, each vector comprises extit{n} real numbers. The dimension extit{n} of this space tells us how many numbers are required to describe each vector fully.

• Closures under addition and scalar multiplication are key features of vector spaces.
• Each operation performed on vectors remains within the space, ensuring stability and structure.

Understanding vector spaces helps in grasping concepts like linear transformations and systems of linear equations. They form the backbone of more complex ideas in linear algebra.

Linear Combinations

Linear combinations are central to understanding vector spaces and subsets. A linear combination involves creating new vectorsby multiplying existing vectors by scalars and then adding the results. This process forms the set called the span of a set of vectors, which includes all possible vectors that can be constructed this way.

Mathematically, if ${\mathbf{v}}_1,{\mathbf{v}}_2,\dots ,{\mathbf{v}}_k$ are vectors in ${\mathbb{R}}^n$, a linear combination is any expression of the form:$$a_1{\mathbf{v}}_1+a_2{\mathbf{v}}_2+\dots +a_k{\mathbf{v}}_k,$$where $a_1,a_2,\dots ,a_k$ are scalars.

This concept is powerful because it not only enables us to generate new vectors but also helps identify relationships and dependencies among vectors.

• The span of a set of vectors is the collection of all their linear combinations, representing all directions and lengths we can explore using these vectors.
• Linear combinations facilitate breaking down complex vector operations into simpler components.

Mastering linear combinations is essential as it leads to further insights into vector independence, basis, and dimension, streamlining vector problem-solving.

Subsets

Subsets play a significant role in linear algebra, especially when considering vector spaces and their spans. A subset is a part of a larger set, where every element in the subset also exists within the larger set. In the context of vector spaces, understanding subsets helps in exploring how one set of vectors is related to another.

When we say that a set of vectors $S$ is a subset of another set $T$, this is mathematically represented as:$$S\subseteq T.$$This means that every vector found in $S$ is also found in $T$. This relationship is crucial when analyzing their spans since the span of a subset is contained entirely within the span of the larger set.

• If $S\subseteq T$, then $\text{span}(S)\subseteq \text{span}(T)$.Thus, vectors form using the linear combinations of $S$ can certainly be formed using $T$.
• Examining subsets makes it easier to understand vector dependency and redundancy.

By recognizing how subsets function within vector spaces, one gains the ability to infer and understand hierarchical relationships between different vector sets.