Question

If \(S\) is a nonempty subset of a vector space \(X\), prove (as asserted in Sec. 9.1) that the span of \(S\) is a vector space.

Short answer

Question: Prove that the span of a nonempty subset S of a vector space X is itself a vector space. Answer: To prove that the span of a nonempty subset S of a vector space X is itself a vector space, we have shown that the span of S satisfies all the vector space axioms, including closure under addition and scalar multiplication, existence of zero vector and additive inverse, as well as properties such as commutativity, associativity of addition, and the distributive property of scalar multiplication, which are inherited from the larger vector space X.

Step-by-step solution

Define the span of S

The span of a nonempty subset S of a vector space X is the set of all linear combinations of the vectors in S. In other words, the span of S is the smallest vector subspace that contains S.

Closure under vector addition

Let v and w be two arbitrary vectors in the span of S. Since they are in the span of S, they can be written as linear combinations of the vectors in S: v = a_1s_1 + a_2s_2 + ... + a_ns_n w = b_1s_1 + b_2s_2 + ... + b_ns_n where s_i are vectors in S and a_i, b_i are scalars. Now consider the vector sum: v + w v + w = (a_1s_1 + a_2s_2 + ... + a_ns_n) + (b_1s_1 + b_2s_2 + ... + b_ns_n) = (a_1 + b_1)s_1 + (a_2 + b_2)s_2 + ... + (a_n + b_n)s_n Since this is again a linear combination of the vectors in S, v + w is also in the span of S. This shows that the span of S is closed under vector addition.

Closure under scalar multiplication

Let v be an arbitrary vector in the span of S and c be a scalar. Since v is in the span of S, it can be written as a linear combination of the vectors in S: v = a_1s_1 + a_2s_2 + ... + a_ns_n Now consider the scalar product: c * v c * v = c * (a_1s_1 + a_2s_2 + ... + a_ns_n) = (c * a_1)s_1 + (c * a_2)s_2 + ... + (c * a_n)s_n Since this is again a linear combination of the vectors in S, c * v is also in the span of S. This shows that the span of S is closed under scalar multiplication.

Existence of zero vector

The zero vector can be represented as a linear combination of the vectors in S with all coefficients being zero: 0 = 0 * s_1 + 0 * s_2 + ... + 0 * s_n Since the zero vector can be written as a linear combination of the vectors in S, it is also in the span of S.

Existence of additive inverse

Let v be an arbitrary vector in the span of S. Since v is in the span of S, it can be written as a linear combination of the vectors in S: v = a_1s_1 + a_2s_2 + ... + a_ns_n Now consider the additive inverse of v, denoted by -v. We can write -v as a linear combination of the vectors in S with the coefficients being the negation of the coefficients of v: -v = (-a_1)s_1 + (-a_2)s_2 + ... + (-a_n)s_n Since -v can be written as a linear combination of the vectors in S, it is also in the span of S.

Commutativity, associativity, and distributive property

Since the span of S is a subset of the vector space X, it inherits the properties of commutativity, associativity of addition, and distributive property of scalar multiplication from the vector space X. Now, we've shown that the span of S satisfies all the vector space axioms. Therefore, the span of a nonempty subset S of a vector space X is indeed a vector space.

Key concepts

Understanding the Span of a Vector Space

The span of a vector space is fundamental to the study of linear algebra. It describes how we can create every possible vector within a space by considering combinations of a small set of vectors from that space. When we say that the span of a nonempty subset, denoted as 'S', within a vector space 'X', is itself a vector space, we mean that 'S' contains all the possible vectors that can be made by scaling and adding its members together.

Imagine having a set of building blocks. Each block represents a vector in 'S'. Combining these blocks in various ways, such as stacking or arranging side by side, represents forming linear combinations. The fact that you can recreate any structure within the confines of 'S' using just these blocks means that the blocks 'span' the space. In mathematical terms, the set of all such possible structures (vectors) you can build is called the 'span of S'.

Exploring Linear Combinations

A linear combination involves taking several vectors and combining them using scalar multiplication and addition. This process is like a recipe that includes ingredients (vectors) and quantities (scalars). Just as you can combine different ingredients in different amounts to create a variety of dishes, you can combine vectors in varying scalar amounts to arrive at new vectors.

For instance, let's consider two vectors 'v' and 'w' represented as:\[v = a_1s_1 + a_2s_2 + \dots + a_ns_n\]\[w = b_1s_1 + b_2s_2 + \dots + b_ns_n\]Here, each \(s_i\) is a 'base vector' from our set 'S', and \(a_i\) and \(b_i\) are the respective scalars. By changing the scalars, we essentially change the 'flavor' of our final vector 'dish'. The ability to form any vector in the span through such combinations confirms that linear combinations are the essence of the span's existence.

Closure Properties: The Glue Holding Vector Spaces Together

The closure properties are like the guarantees in a vector space 'recipe book', ensuring that whatever vectors you cook up using linear combinations, they will still belong to the same 'cuisine', or in mathematical terms, to the same vector space. Closure under addition and scalar multiplication are two vital aspects that any set must satisfy to be called a vector space. These properties ensure that when we perform vector addition or scalar multiplication, the result doesn't 'escape' the space.

As illustrated in the textbook example, by showing the addition of two vectors in the span results in another vector in the span and that multiplying a vector in the span by a scalar yields yet another vector that still lies within the span, we've effectively 'closed' the span under these operations. This demonstrates a fundamental concept in vector spaces: no matter how we mix and remix vectors through addition or stretching through scalar multiplication, we stay within the boundaries of the vector space defined by the span of 'S'.