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The concept of span is essential in understanding how vectors relate to each other in a vector space. When we talk about the span of vectors, we are referring to all possible linear combinations that can be formed using a given set of vectors.
Let's break this down a bit more. Suppose you have a set of vectors, such as \( \operatorname{span}(\mathbf{a}_1, \mathbf{a}_2, \ldots, \mathbf{a}_n) \). The span includes any vector that can be expressed as:

• \( c_1 \mathbf{a}_1 + c_2 \mathbf{a}_2 + \ldots + c_n \mathbf{a}_n \) where \( c_1, c_2, \ldots, c_n \) are scalars.

In the given exercise, we examine the span of \( \mathbf{u}, \mathbf{u} + \mathbf{v}, \mathbf{u} + \mathbf{v} + \mathbf{w} \). Understanding this span means recognizing that any vector created from these building blocks can be a combination of these three specific vectors.
This enables us to express seemingly different vectors, namely \( \mathbf{u}, \mathbf{v}, \) and \( \mathbf{w} \), as parts of this span, as demonstrated in the manipulation of these vectors in the solution.

Vector spaces are fundamental constructs in linear algebra. They include vectors, which can be added together and multiplied by scalars to create new vectors within the same space. This means that vector spaces adhere to specific rules and properties that guarantee the arithmetic operations of addition and scalar multiplication are well-defined.
Properties of vector spaces include:

• Closure under addition and scalar multiplication: The result of adding two vectors or multiplying a vector by a scalar is another vector in the same space.
• Existence of a zero vector: There is a vector of zero magnitude (or length) in the set.
• Existence of additive inverses: For every vector, there is another vector such that the sum of the two is the zero vector.

The exercise exemplifies a subspace of the vector space, more precisely, a span of vectors that is closed under linear combinations. This closure is crucial because it allows us to express \( \mathbf{u}, \mathbf{v}, \) and \( \mathbf{w} \) within the defined vector space, using a step-by-step process to show these vectors belong to the span.

In mathematics, proving statements or hypotheses requires specific techniques to show a claim is valid. Proofs are the logical deductions that confirm the truth of a statement, often involving a sequence of established steps.
The original exercise involves a proof to show that \( \mathbf{u}, \mathbf{v}, \) and \( \mathbf{w} \) can be expressed as linear combinations of \( \mathbf{u}, \mathbf{u} + \mathbf{v}, \) and \( \mathbf{u} + \mathbf{v} + \mathbf{w} \).

• First Technique: Direct computation - This begins with manipulating terms to form desired results; in this exercise, direct calculation was applied by breaking down each vector using arithmetic operations.
• Stepwise verification - This whether each step logically follows the last, often requiring breaking down complex expressions into simpler ones.

Using these strategies, we craft an understanding that confirms \( \mathbf{u}, \mathbf{v}, \) and \( \mathbf{w} \) lie within the span. By expanding out the vectors into their component forms, we ensure that every part of the vector lies in the span, supporting the overall structure of the proof.