91Ó°ÊÓ

In linear algebra, a fundamental concept is the linear combination. Essentially, a linear combination of a set of vectors is formed by multiplying each vector by a scalar and then summing the results. This is crucial because it helps us determine how vectors relate to each other within a set. Suppose you have vectors \(\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_k\) in a vector space \(\mathbb{R}^n\). A vector \(\mathbf{v}\) is a linear combination of these vectors if there exist scalars \(c_1, c_2, \ldots, c_k\) such that:

• \(\mathbf{v} = c_1 \mathbf{v}_1 + c_2 \mathbf{v}_2 + \cdots + c_k \mathbf{v}_k\).

This expression illustrates that \(\mathbf{v}\) is built from the set of vectors using specific scalar multipliers, showcasing the importance of each vector in forming another.

A vector space is a mathematical structure formed by a collection of vectors, which can be added together and multiplied by scalars. These operations comply with certain rules such as commutativity, associativity, and distributivity. The set \(\mathbb{R}^n\) is a classic example of a vector space, where each vector has \(n\) dimensions or components.
Here are key characteristics of vector spaces:

• Closed under addition and scalar multiplication.
• Contains the zero vector serving as an additive identity.
• Allows for the creation of linear combinations.

Understanding vector spaces is critical because they provide the framework where concepts like linear combinations, bases, and dimensions actively operate.

The span of a set of vectors is the collection of all possible linear combinations of those vectors. If you have a set \(S = \{\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_k\}\), the span, \(\operatorname{span}(S)\), includes every vector that can be expressed as:

• \(c_1 \mathbf{v}_1 + c_2 \mathbf{v}_2 + \cdots + c_k \mathbf{v}_k\), where \(c_i\) are real numbers.

The concept of span is important because it tells us how large a vector space can be generated from a set of vectors.
In the given exercise, proving that \(\operatorname{span}(S)\) is equal to \(\operatorname{span}(S')\) involves showing that adding or removing a vector that is a linear combination of others doesn’t change the overall span.

Proof writing is a critical skill in mathematics that involves presenting a logical and systematic argument to establish the truth of a statement. It requires a careful arrangement of logical steps supported by definitions, theorems, and previously established results.
In our exercise, we demonstrated that \(\operatorname{span}(S) = \operatorname{span}(S')\) by proving both:

• \(\operatorname{span}(S') \subseteq \operatorname{span}(S)\)
• \(\operatorname{span}(S) \subseteq \operatorname{span}(S')\)

By completing these steps, we conclude that the spans are equal, emphasizing clarity and structure in proof writing. Mastering this enables deeper understanding and communication of mathematical ideas, essential for solving a broad array of problems in linear algebra and beyond.