26.4. Linear Combination and Span

A linear combination of vectors v₁, v₂, ..., vₖ ∈ V is any vector of the form a₁v₁ + a₂v₂ + ... + aₖvₖ, where aᵢ ∈ 𝔽.

A linear combination involves taking several vectors — denoted as v₁, v₂, ..., vₖ — and creating a new vector by scaling each one by a corresponding scalar, which we denote as a₁, a₂, ..., aₖ. The symbols aᵢ represent real numbers (or complex numbers) from the field 𝔽. This means you multiply each vector by a number and then sum the results. For example, if v₁ = (1, 2) and v₂ = (3, 4), and we take a₁ = 2 and a₂ = 3, the linear combination would be 2*(1, 2) + 3*(3, 4) = (2, 4) + (9, 12) = (11, 16).

Think of making a smoothie. Each type of fruit you add (like bananas and strawberries) represents a vector. The amount of each fruit (like 2 bananas and 3 strawberries) represents the scalar values. When you blend them (adding the scaled versions of each fruit), you create a new delicious smoothie (the resulting vector).

The span of a set S = {v₁, v₂, ..., vₖ} ⊆ V is the set of all linear combinations of vectors in S. Span(S) = {∑ aᵢvᵢ | aᵢ ∈ 𝔽}

The span of a set of vectors is the collection of all possible vectors you can create by taking linear combinations of those vectors. If you have a set S of vectors (v₁, v₂,...,vₖ), then the span of S includes every vector that can be made by scaling each vector in S and adding them together. So, for every combination of scalars a₁, a₂, ..., aₖ from the field 𝔽, the resulting vector belongs to the span.

Imagine you have various colors of paint, like red and blue. By mixing different amounts of each color, you can create new colors (like purple). The span of your colors represents all possible colors you could create using just red and blue paint, taking into account that you can vary the amounts.

Span(S) is always a subspace of V.

A subspace is a subset of a vector space that is also a vector space under the same operations (vector addition and scalar multiplication). The set of vectors formed by taking all the linear combinations of a set S is not just any collection, but it holds to the rules of vector spaces; therefore, Span(S) is indeed a subspace of the larger vector space V.

Consider a large garden (the vector space V) with various types of plants. If you only consider the tulips you have planted (the set S), all the possible arrangements and combinations of those tulips (like different groupings or clusters) exist within the context of the larger garden — they form their own little garden space, which is the span.

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