How can you determine if a set of given vectors can span a subspace in R^n?

  • Thread starter mckallin
  • Start date
  • Tags
    Set
In summary: The vectors cannot span R^4 because the columns of the matrix formed from the vectors do not satisfy the equation:[1 3 -5 0]=-[2 1 0 0] [0 2 1 -1] [1 -4 5 0]
  • #1
mckallin
15
0
support that p is in M and M=span{v,w},that means p=av+bw, right?
then, could either a or b be zero?

regarded to this following problem:
Is is possible that {[1 2 0]^T,[2 0 3]^T} can span the subspace U={[r s 0]^T / r and s in R}?


I thought if s=2r and the scalar for [2 0 3]^T could be zero, it might be possible?
 
Physics news on Phys.org
  • #2
U is a two dimensional vector space so [r 2r 0]^T is only a one dimemsional subspace of it. [1 2 0]^T spans that subspace of U but not U.

The question is, "suppose [r s 0]^T is a general vector in U. Do there exist numbers x, y such that [r s 0]^T= x[1 2 0]^T+ y[2 0 3]^T?" If so then we would have [r s 0]^T= [x 2x 0]^T+ [2y 0 3y]= [x+ 2y 2x 3y]. The means we must have r= x+ 2y, s= 2x, 0= 3y. From the last equation, obviously y= 0. But then we r= x, s= 2x. Since r and s can be any two real numbers, there does not exist a number x satifying both equations.
 
  • #3
I see, thx.
how about my first question, I would still like to know the answer to that:
support that p is in M and M=span{v,w},that means p=av+bw, right?
then, could either a or b be zero?Also, I have just seen a problem that asks for a spanning set for the zero subspace {0} of R^n.
Then, I let [0 0 0 0]^T be the zero subspace of R^4, so is this a spanning set for that zero subspace:
span{[0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0]}
 
Last edited:
  • #4
mckallin said:
I see, thx.
how about my first question, I would still like to know the answer to that:
support that p is in M and M=span{v,w},that means p=av+bw, right?
Yes.

then, could either a or b be zero?
Yes.


Also, I have just seen a problem that asks for a spanning set for the zero subspace {0} of R^n.
Then, I let [0 0 0 0]^T be the zero subspace of R^4, so is this a spanning set for that zero subspace:
span{[0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0]}
Why do you have the same thing four times in a set? This set is exactly the same thing as {[0 0 0 0]}, which is a spanning set {0}= {[0 0 0 0]}
 
  • #5
thx.
I am sorry that I have got a more problem right away. Could you help again?

Do you know how to determine if a set of some given vectors in R^n can span R^n?
regarded to this question:

Determine if the following given vectors span R^4.
{[1 3 -5 0] [-2 1 0 0] [0 2 1 -1] [1 -4 5 0]}.


I thought they could because I seem to be able to reduce the matrix with these vectors as columns into an identity matrix. But the answer in the book says they can't. I was getting confused.
 

Related to How can you determine if a set of given vectors can span a subspace in R^n?

What is a spanning set?

A spanning set is a set of vectors that can be used to create any other vector in a vector space through linear combinations. In other words, a spanning set contains enough vectors to "span" the entire vector space.

Why is a spanning set important in linear algebra?

A spanning set is important because it allows us to determine the dimension of a vector space. Additionally, it helps us understand the structure of a vector space and how to perform operations on vectors within that space.

How do you determine if a set of vectors is a spanning set?

To determine if a set of vectors is a spanning set, we need to check if every vector in the vector space can be written as a linear combination of the vectors in the set. If this is true, then the set is a spanning set. If not, then the set is not a spanning set.

Can a spanning set be infinite?

Yes, a spanning set can be infinite. As long as the set contains enough vectors to span the entire vector space, it can be considered a spanning set.

What is the difference between a basis and a spanning set?

A basis is a special type of spanning set that is linearly independent, meaning that none of the vectors in the basis can be written as a linear combination of the other vectors. A basis is the smallest possible spanning set for a vector space and is unique for each vector space.

Similar threads

  • Linear and Abstract Algebra
Replies
6
Views
1K
  • Linear and Abstract Algebra
Replies
19
Views
4K
Replies
2
Views
843
  • Linear and Abstract Algebra
Replies
3
Views
1K
  • Linear and Abstract Algebra
Replies
3
Views
2K
  • Linear and Abstract Algebra
Replies
1
Views
1K
  • Linear and Abstract Algebra
Replies
1
Views
861
  • Linear and Abstract Algebra
Replies
5
Views
1K
  • Linear and Abstract Algebra
2
Replies
39
Views
2K
  • Linear and Abstract Algebra
Replies
1
Views
1K
Back
Top