# how do vector spaces relate to basis

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?

## 3 Answers

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


Definiton of a vector space, has three parts:

1- an abelian group $$G$$, with its operator(usually written by +) which in your case is $$\mathbb R^2$$ and usual addition of vectors. The elements of this group are called vectors.

2- a base field $$F$$, which in your case is $$\mathbb R$$(set of real numbers with its usual addition and multiplication).

3- a scalar product which is a function from $$F×G$$ to $$G$$. This scalar product function must satisfy some axiomswhich you can find them here in definition part(given in a table)

Now, after getting a vector space, for any set of vectors $$V\subset G$$ two important concepts are defined :

1. Span of $$V$$(which is defined by Daniel)
2. Linear dependance of $$V$$(which you can find its definition here)

We know that for any vector space $$G$$ there is a set of vectors which both of these concepts are satisfied(it spaans $$G$$ and is linear independent). This is because of axiom of choice which is a set theoritic principle. These sets are called "Basis of vector space" and they are not uniqe. For example, in your case, {(0,1) , (1,0)} is a basis. But, not every set of vectors form a basis. There are set of vectors which dosn't span the space(their span set is smaller than whole space) and there are set of vectors which are linear dependent.

So we see that span, is a concept realted to set of vectors, defined in vector space.

The concept of span is essential to the definition of a basis, but not to a vector space.

As Sim000 points out, a vector space $$V$$ consists of, in simple terms, a way of adding and subtracting elements of a set (which we call vectors) and multiplying ("scaling") them by elements of some field $$K$$ (where a field behaves like the rational numbers equipped with addition, subtraction, multiplication and division). We usually study the cases when $$K$$ is either the real numbers or the complex numbers.

Given the definition of a vector space, the span of a family $$\lbrace v_1,\dots,v_n \rbrace$$ of vectors is the set of all linear combinations of those vectors. In your example, you're restricting yourself to the span of a pair of linearly independent vectors in $$\mathbb{R}^2$$, and it's a simple fact of linear algebra that $$n$$ linearly independent vectors in $$\mathbb{R}^n$$ will span $$\mathbb{R}^n$$.

We define a basis $$\mathcal{B}$$ of a vector space $$V$$ to be any set of linearly independent vectors whose span is all of $$V$$. It's a simple fact that any such set will always have the same number of vectors, and we call this number the dimension of $$V$$.

As Daniel points out in his answer, you can in fact work the other way around, and use the concept of span to define linear subspaces and in turn vector spaces.