Problem 48 - Differential Equations and Linear Algebra

Question

Prove that if \(S\) and \(S^{\prime}\) are subsets of a vector space \(V\) such that \(S\) is a subset of \(S^{\prime},\) then \(\operatorname{span}(S)\) is a subset of \(\operatorname{span}\left(S^{\prime}\right)\)

Step-by-Step Solution

Answer

To prove that if $S \subseteq S'$, then $\operatorname{span}(S) \subseteq \operatorname{span}\left(S'\right)$, take a vector \(x \in \operatorname{span}(S)\), which can be written as a linear combination of vectors in S: \(x = \sum_{i=1}^k c_i v_i, v_i \in S\), and \(c_i \in F\). Since \(S \subseteq S'\), we have \(v_i \in S'\) and the same linear combination can be written with vectors in \(S'\), confirming that \(x\) also belongs to \(\operatorname{span}\left(S'\right)\). Thus, \(\operatorname{span}(S) \subseteq \operatorname{span}\left(S'\right)\).

Step 1: Define the span of the subsets

The span of a vector space is the set of all linear combinations of the vectors in that space. So, for any subset T of vector space V, we can define its span as: \[\operatorname{span}(T) = \left\{x \in V: x=\sum_{i=1}^k c_i v_i, k \ge 0, c_i \in F, v_i \in T\right\}\] where F is the field over which the vector space V is defined. Now, we need to prove that if a vector x belongs to the span of S, it also belongs to the span of S'.

Step 2: Write a vector in span of S as a linear combination of vectors in S

Let's take a vector x belonging to the span of S. By definition of the span, we can write x as a linear combination of vectors in S: \[x = c_1 v_1 + c_2 v_2 + \dots + c_k v_k, \text{ where } v_i \in S \text{ and } c_i \in F.\]

Step 3: Use the fact that S is a subset of S'

Since S is a subset of S', every vector in S also belongs to S'. So, each \(v_i\) in the linear combination above can be considered to belong to S': \[x = c_1 v_1 + c_2 v_2 + \dots + c_k v_k, \text{ where } v_i \in S' \text{ and } c_i \in F.\]

Step 4: Conclude that the vector belongs to the span of S'

Now, since the vector x can be written as a linear combination of vectors in S', it belongs to the span of S'. \[x \in \operatorname{span}\left(S'\right)\]

Step 5: Confirm that span(S) is a subset of span(S')

We have shown that for any vector x belonging to the span of S, x also belongs to the span of S'. Therefore, we conclude that the span of S is a subset of the span of S': \[\operatorname{span}(S) \subseteq \operatorname{span}\left(S'\right)\]

Key Concepts

Subsets of Vector Space

Understanding subsets within a vector space is crucial when studying linear algebra. A vector space, to remind you, is a collection of vectors that can be added together and multiplied by scalars, following certain rules.

Subsets of a vector space are simply collections of vectors from that vector space that may or may not form a vector space themselves. For instance, if you consider the entire 3-dimensional space, a plane or a line within that space could be considered subsets.

Importantly, when we focus on the span of such subsets, we're looking at all possible linear combinations that can be created from vectors within that subset. If a subset 'A' is contained within another subset 'B', intuitively, the span of 'A' should also be contained within the span of 'B', as every vector in 'A' is also a part of 'B' and therefore any combination of vectors from 'A' can also be assembled using vectors from 'B'.

Linear Combination

A linear combination in a vector space involves constructing new vectors by multiplying existing vectors by scalars and adding the results together. For example, if 'a' and 'b' are vectors, then any vector of the form \(ca + db\), where 'c' and 'd' are scalars, is a linear combination of 'a' and 'b'.

This process is the building block of many operations in vector spaces, such as defining the span or solving systems of linear equations. In the given problem, we showed that any vector in the span of a subset 'S' is a linear combination of the vectors in 'S'. Since 'S' is entirely contained in a larger subset 'S''', this means the same linear combinations that you can build with 'S' can also be constructed with 'S''' — making the span of 'S' naturally a subset of the span of 'S'''.

Vector Space Properties

Vector spaces have specific properties that must be satisfied for any collection of vectors to be considered a vector space. These include closure under addition and scalar multiplication, the existence of a zero vector, and the ability for each vector to have an additive inverse.

Within vector spaces, subsets that themselves satisfy all vector space properties are called 'subspaces'. To illustrate, while any line through the origin in a 3D space is a subspace, not every subset, like a single point or a curve that isn't straight, qualifies as one.

In our exercise solution, understanding vector space properties helps us see why the span of any set of vectors forms a subspace. It confirms why the span of a smaller subset 'S' is also contained in the span of a larger subset 'S''', relating closely to the property of closure under addition and scalar multiplication, which guarantees that all linear combinations remain within the vector space.