Subspace & Span in Rn: Definition & Examples

  • Thread starter nobahar
  • Start date
  • Tags
    Subspaces
In summary, the conversation discusses the concept of span and its relationship to a subspace in \mathbb{R}^n. The question is raised about whether \mathbb{R}^n can be considered a subspace of itself, which is confirmed to be true. The example given further clarifies this concept.
  • #1
nobahar
497
2
Hello!
Just a quick question. Is the following okay?:
The span of a set of vectors corresponds to a subspace in Rn.
But the span of a set of vectors can also be ALL of Rn, does that mean all of Rn can be considered a subspace? Or does it mean the first definiton is not entirely correct, and instead a span of a set of vectors can simply be used to identify a subspace, if it exists.
I realize this is a fairly trivial question, but I would like to make sure that I haven't overlooked anything.
Here is an example of what I mean.
The span(v1,v2), where [tex]v_{1} = \left(\begin{array}{cc}1\\0\end{array}\right), v_{2} = \left(\begin{array}{cc}0\\1\end{array}\right)[/tex] is R2. It is also closed under addition and multiplication, and contains the zero vector, so it satisfies the requirements for a subspace; is it a subspace within, say Rn with n>2? I don't think so, because the vector is composed of two components, and I was thinking a subspace would have to be within the confines of these two dimensions.
Any help appreciated, if it is not clear I can try to re-explain what I mean. I am hoping it's more of an issue of definition.
Thanks in advance.
 
Physics news on Phys.org
  • #2
nobahar said:
Hello!
Just a quick question. Is the following okay?:
The span of a set of vectors corresponds to a subspace in Rn.
But the span of a set of vectors can also be ALL of Rn, does that mean all of Rn can be considered a subspace?
Yes, a vector space is a subspace of itself.
 
  • #3
[itex]\mathbb{R}^n[/itex] is a subspace of itself (you can verify with the definition that it satisfies the properties of a subspace).
 
  • #4
Thankyou Vela and Rasmhop.
 

1. What is a subspace in Rn?

A subspace in Rn is a subset of the vector space Rn that satisfies three conditions: it contains the zero vector, it is closed under vector addition, and it is closed under scalar multiplication.

2. What is the definition of span in Rn?

The span of a set of vectors in Rn is the set of all possible linear combinations of those vectors. In other words, it is the set of all vectors that can be created by multiplying each vector by a scalar and then adding them together.

3. How do you determine if a set of vectors spans a subspace in Rn?

To determine if a set of vectors spans a subspace in Rn, you can use the span definition and see if all possible linear combinations of the vectors are contained within the subspace. If they are, then the vectors span the subspace.

4. Can a subspace in Rn have an infinite number of dimensions?

Yes, a subspace in Rn can have an infinite number of dimensions. This means that the number of vectors needed to span the subspace is infinite.

5. How is the concept of subspace and span used in real-world applications?

The concept of subspace and span is used in a variety of fields, such as engineering, physics, and computer science. It is used to model and analyze systems that have multiple variables and can be represented as vectors. For example, in computer graphics, subspace and span are used to create 3D models and animations by manipulating vectors in Rn.

Similar threads

  • Precalculus Mathematics Homework Help
Replies
4
Views
2K
  • Linear and Abstract Algebra
Replies
8
Views
731
  • Calculus and Beyond Homework Help
Replies
4
Views
1K
  • Linear and Abstract Algebra
Replies
6
Views
710
  • Precalculus Mathematics Homework Help
Replies
6
Views
1K
  • Precalculus Mathematics Homework Help
Replies
12
Views
2K
  • Calculus and Beyond Homework Help
Replies
15
Views
2K
  • Precalculus Mathematics Homework Help
Replies
2
Views
2K
  • Quantum Physics
Replies
12
Views
2K
  • Linear and Abstract Algebra
Replies
3
Views
780
Back
Top