Question

We often write vectors in \(\mathbb{R}^{n}\) as rows. Suppose that \(\mathbf{x}_{1}, \mathbf{x}_{2}, \ldots, \mathbf{x}_{k}\) are vectors in \(\mathbb{R}^{n} .\) If \(\mathbf{y}=a_{1} \mathbf{x}_{1}+a_{2} \mathbf{x}_{2}+\cdots+a_{k} \mathbf{x}_{k}\) where \(a_{1} \neq 0,\) show that \(\operatorname{span}\left\\{\mathbf{x}_{1} \mathbf{x}_{2}, \ldots, \mathbf{x}_{k}\right\\}=\operatorname{span}\left\\{\mathbf{y}_{1}, \mathbf{x}_{2}, \ldots, \mathbf{x}_{k}\right\\}\)

Short answer

The spans are equal because each vector in one set is a combination of vectors in the other set.

Step-by-step solution

Understanding the Problem

We need to show that the span of the set of vectors \(\{\mathbf{x}_{1}, \mathbf{x}_{2}, \ldots, \mathbf{x}_{k}\}\) is the same as the span of the set \(\{\mathbf{y}, \mathbf{x}_{2}, \ldots, \mathbf{x}_{k}\}\), where \(\mathbf{y}=a_{1} \mathbf{x}_{1}+a_{2} \mathbf{x}_{2}+\cdots+a_{k} \mathbf{x}_{k}\) and \(a_{1} eq 0\).

Expressing y in Terms of x_1

Since \(\mathbf{y}=a_{1} \mathbf{x}_{1}+a_{2} \mathbf{x}_{2}+\cdots+a_{k} \mathbf{x}_{k}\), we can solve for \(\mathbf{x}_{1}\) in terms of \(\mathbf{y}, \mathbf{x}_{2}, \ldots, \mathbf{x}_{k}\): \[\mathbf{x}_{1} = \frac{1}{a_{1}} \left( \mathbf{y} - a_{2} \mathbf{x}_{2} - \cdots - a_{k} \mathbf{x}_{k} \right)\] because \(a_{1} eq 0\).

Proving Inclusion of Spans

We will show that each vector set is included in the span of the other. First, every vector in \(\operatorname{span} \{\mathbf{x}_{1}, \mathbf{x}_{2}, \ldots, \mathbf{x}_{k}\}\) is already in \(\operatorname{span} \{\mathbf{y}, \mathbf{x}_{2}, \ldots, \mathbf{x}_{k}\}\) since \(\mathbf{x}_{1}\) can be expressed as a combination of these vectors, and the other vectors (\(\mathbf{x}_{2}, \ldots, \mathbf{x}_{k}\)) are included. Conversely, the vector \(\mathbf{y}\) is a linear combination of the vectors in the first span set.

Conclusion and Verification

Both spans are equivalent because each element from one set can be expressed as a linear combination of the elements of the other set. Thus, \(\operatorname{span}\{\mathbf{x}_{1}, \mathbf{x}_{2}, \ldots, \mathbf{x}_{k}\} = \operatorname{span}\{\mathbf{y}, \mathbf{x}_{2}, \ldots, \mathbf{x}_{k}\}\).

Key concepts

Linear Combinations

A linear combination in the world of vector spaces involves taking several vectors and combining them using scalars. Consider vectors \(\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_k\) in a vector space, where each vector can be multiplied by a scalar. You can then sum these products to form a new vector. For instance: \(\mathbf{y} = a_1 \mathbf{x}_1 + a_2 \mathbf{x}_2 + \cdots + a_k \mathbf{x}_k\). Each \(a_i\) is a scalar, and if \(a_1 eq 0\), we can say that \(\mathbf{x}_1\) substantially contributes to forming \(\mathbf{y}\).

• Scalars can be any real or complex number.
• The vectors are the building blocks, while the scalars determine the weight of each vector in the combination.
• Linear combinations are fundamental in defining spans, as they help in generating new vectors from existing ones.

Understanding linear combinations is crucial, as they reveal how change in one vector affects the whole structure of a span or vector space.

Linear Algebra

Linear algebra forms the backbone of vector spaces and linear combinations. It deals with matrices, vectors, and operations like addition and multiplication on these structures. This branch of mathematics provides tools to solve systems of linear equations and to understand transformations in space.

• Vector spaces: A set of elements (vectors) where vector addition and scalar multiplication are defined and satisfy certain properties.
• Matrices: Arrays of numbers that can represent transformations, such as rotations and reflections in space.
• Linear transformations: Functions that map vectors to other vectors, preserving the structure of the space.

Linear algebra simplifies complex problems in various fields such as physics, computer science, and economics by translating them into functions and systems that are easier to analyze and solve.

Vector Span

The concept of a vector span introduces the range of possibilities you can reach with linear combinations of a set of vectors. It comprises all the linear combinations possible using a given set of vectors. For a set of vectors \(\{\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_k\}\), any vector within the span can be expressed as a linear combination of these vectors. In our case of vectors in \(\mathbb{R}^n\), this idea forms the foundation for subspaces.

• The span of a vector set is essentially a plane or space that encompasses all possible vectors that can be formed from the original set.
• A span can be limited by its vectors; adding a non-zero vector to a span can expand it or change its dimension.
• Spans are useful in determining the coverage of a vector space or finding solutions to vector equations.

Recognizing how spans work helps in visualizing solutions to various problems in linear algebra, ensuring that transformations and other operations maintain consistency within the same dimensional framework.