[Finite-Dimensional Linear Algebra] 2.4 Linear Combinations and Spanning Sets

Definition of a Linear Combination

Let `V` be a vector space over a field `F`, let `u_1, u_2,\dots u_k` be vectors in `V`, and let `\alpha _1, \alpha _2,\dots \alpha _k` be scalars in `F`. Then the linear combination of the vectors with scalars we call weights is: ``\alpha _1 u_1+\alpha _2u_2+\cdots + \alpha _ku_k``

Any linear combination of vectors in subspace `S` belongs to `S`, as subspaces are closed under the operations of addition and scalar multiplication. Subspaces are often defined or represented by linear combinations. 

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Definition of a Spanning Set

Let `V` be a vector space over a field `F`, let `u_1, u_2,\dots u_n` be vectors in `V`, where `n \geq 1`. Let `S` be the set of all linear combinations of `u_1, u_2,\dots u_n`: ``S=\{ \alpha _1u_1 + \alpha _2 u_2 + \dots + \alpha _nu_n : \alpha _1, \alpha _2, \dots , \alpha _n \in F\}``

Then `S` is a subspace of `V`, and it is called the span of `u_1, u_2,\dots u_n` and is denoted by `S = sp \{ u_1, u_2,\dots u_n \}`.

• the set of vectors { `u_1, u_2,\dots u_n` } is a spanning set for `S`
• `u_1, u_2,\dots u_n` span `S`

Figure 2.4.1. Span of a single vector in 2D space

Every subspace is a span of some set of vectors, and every span is a subspace. 

 

Q. Given a subspace `S` of a vector space `V` and a vector `v \in V`, does `v` belong to `S`? 

A. If `S = sp \{ u_1, u_2,\dots u_n \} `, and `v` can be written as a linear combination of ` u_1, u_2,\dots u_n`, then `v` belongs to `S` 

 

Q. How does one show a set of vectors spans a vector space `V`?

A. One can either

1. show that every vector in V can be written as a linear combination of the set of vectors 
2. for finite-dimensional space, show that it has as many vectors as the dimension of the space, and they are linearly independent

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spanning set: finite representation of a finite-dimensional vector space or subspace