Dimension of the span of a set of vectors

  • #1

Main Question or Discussion Point

My linear algebra is a bit rusty.

Let ##A=\{\bar{v}_1, \dots, \bar{v}_1\}## be a set of vectors in ##R^n##. Can dim(span##(A))=n## without spanning ##R^n##?

I guess I'm unclear on how to interpret the dimension of the span of a set of vectors.
 

Answers and Replies

  • #2
andrewkirk
Science Advisor
Homework Helper
Insights Author
Gold Member
3,788
1,386
The span of a set of vectors is a vector space. There cannot be a proper n-dimensional subspace of an n-dimensional vector space. Any n-dimensional subspace must be the whole thing.

That is one area where vector spaces differ from modules.
 
  • #3
33,085
4,792
My linear algebra is a bit rusty.

Let ##A=\{\bar{v}_1, \dots, \bar{v}_1\}## be a set of vectors in ##R^n##. Can dim(span##(A))=n## without spanning ##R^n##?

I guess I'm unclear on how to interpret the dimension of the span of a set of vectors.
Are there n vectors in A? Your subscripts all appear to be 1. Presumably you meant ##\{\bar{v}_1, \dots, \bar{v}_n\}##.
 
  • #4
Are there n vectors in A? Your subscripts all appear to be 1. Presumably you meant ##\{\bar{v}_1, \dots, \bar{v}_n\}##.
Yeah that second, 1 should have been an m. Not necessarily an n. (at least I don't think it should necessarily be an n).
 
  • #5
The span of a set of vectors is a vector space. There cannot be a proper n-dimensional subspace of an n-dimensional vector space. Any n-dimensional subspace must be the whole thing.

That is one area where vector spaces differ from modules.
Ok, this is exactly what I needed to know!

I've not gotten to modules yet. I'm taking a second graduate Algebra class next semester though. I'm told we will finally get into them then.
 
  • #6
33,085
4,792
Yeah that second, 1 should have been an m. Not necessarily an n. (at least I don't think it should necessarily be an n).
Then you should specify some condition on m. If m < n, then your set of vectors could not possibly span ##\mathbb{R}^n##, since dim(span(A)) ##\le m < n##. If ##m \ge n##, the set of vectors might span ##\mathbb{R}^n##, or might not.
 
  • #7
The only condition I wanted on m is that it was not necessarily equal to n. So that it was truly an arbitrary set of vectors in ##R^n##. That way the question was just: "Given an arbitrary set of vectors, A, such that dim(span(A)) = n. Does A necessarily span ##R^n##?" This is a more concise way of asking the question.. I believe the two questions are equivalent though. If not, this is definitely the question I mean to ask.

Which the answer appears to be yes... as per andrewkirk.

Edit: I guess the condition that dim(span(A))=n forces the condition that ##m\geq n##.
 
  • #8
33,085
4,792
Edit: I guess the condition that dim(span(A))=n forces the condition that m n .
Yes
 

Related Threads for: Dimension of the span of a set of vectors

  • Last Post
Replies
21
Views
2K
Replies
2
Views
1K
Replies
5
Views
645
  • Last Post
Replies
3
Views
3K
Replies
2
Views
2K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
6
Views
10K
Replies
5
Views
10K
Top