In this post, we consider “span” of a vector space . As we know, definition of span is:
Definition 1 Let be a finite set of vectors in a vector space . The subset of spanned by is the set of all linear combinations of . This set is called the span of and is denoted
For example, you can see
Example 1 Let , we have . Of course, . It is easy to see that because . That system has always solution . This implies .
We have the following result
Theorem 2 The span of any finite set of vectors in a vector space is a subspace of .
- Suppose is a subspace of a vector space . Prove that if , then .
- Suppose and are vectors in a vector space satisfy:
if and then .
R. Messer, Linear algebra gateway to mathematics, Harper Collins College Publishers
D. Lay, Linear algebra and its applications, Addison-Wesley, 2012.