Answer to Question #143192 in Linear Algebra for Ojugbele Daniel

Question

Let v be a vector space over F, define a spanning of v.

Answer 1

First of all, if the original question is exactly how you have written it, the answer is trivial : the span of a vector space is the vector space itself. If it is not the case and you want a definition of a span of a vector, a set of vectors, or just a span of any subset $X \subset V$ , here it is

We can define $Span(X)$ for $X \subset V$ by two different manners:

1. More abstract, but more universal and useful : $Span(X) = \underset{W-\text{vector subspace of }V, X\subset W}{\cap} W$ . You take juste an intersection of all the linear (vector) subspaces of V containing X. For this definition to work you need to proof that an intersection of vector spaces is still a vector space, which is not that hard.
2. More straightforward, you define $Span(X)$ by all possible finite (!!! it is important) linear combinations of vectors in X: $Span(X)=\{\sum_{i=1}^k \lambda_iv_i, v_i\in X, \lambda_i\in F\}$ . You can verify that it is indeed a vector space.

You can even proof that two definitions are equivalent and therefore you will have 2 different characterisations of $Span(X)$ : explicit formula approach (2) and the smallest space that contains X approach (1).

Remark : in the first definition i have considered only the case $Span(X), X\subset V$ , as other cases (a vector or a set of vectors) are just particular cases of this.

Remark 2 : if $Span(X)=V$ , we say that $X$ generates/spans/is a spanning set of $V$ .