How do vector spaces relate to basis

linear algebravector-spacesvectors

My understanding is that a vector space is defined by the span of two linearly independent vectors. Any two linearly independent vectors that can define the vector space can be said to be the basis for the vector space. For example the vector (1,0) and (0,1) are the basis for the Euclidean plane but using (2,0) and (0,2) would be just as valid a basis for the Euclidean plane, just maybe less convenient. Is this the correct conceptual relationship?

If yes, then what is the difference between span and vector space?

Best Answer

Yes, you are (mostly) correct. Vector spaces can be defined by the span of it's linearly independent vectors (it's basis).

Note: The term "vector space" may also be defined by other means, which I am assuming you will discover later in your studies of linear algebra.

The terms span and vector space are not purely synonymous. If you take the span of a few linearly independent vectors, it won't always produce (or fill) the entire vector space.

Take for example:

Span{(1,0,0),(0,1,0)} will only produce the, as you mentioned, Euclidean plane ($\mathbb{R}^{2}$), even though the entire vector space under observation is $\mathbb{R}^{3}$.

More formally a definition for the Span:

The set of all linear combinations of a list of vectors $v_{1},...,v_{m}$ in vector space $\mathbb{V}$ is called the span of $v_{1},...,v_{m}$, denoted span($v_{1},...,v_{m}$). In other words,

span($v_{1},...,v_{m}$)={$a_{1}v_{1} + \cdots + a_{m}v_{m}:a_{1},...,a_{m} \in \mathbb{R}$} 

                      Axler. S, 2015 p29, Linear Algebra Done Right
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