Linear Algebra ~ Method to determine if a sequence of vectors is a basis

In summary: R^n?No, the set of vectors can always be spanned in R^n. It's just that if you have too few vectors, then the set is not linearly independent and it's not a basis. Conversely, if you have too many vectors, then the set is linearly dependent and it's not a basis.
  • #1
number0
104
0

Homework Statement



A lot of my homework asks me to determine if a given matrix (sequence of vectors) is a basis or not.


Homework Equations





The Attempt at a Solution



Can I just find the reduced echelon form of a given matrix and see if it is linear independent or linear dependent. If it is linear independent, then it is a basis. Otherwise, it is not.

Is this acceptable? Thanks.
 
Physics news on Phys.org
  • #2
Yes, because by definition every element of a basis is linearly independent of the others.
 
  • #3
For a set of vectors to be a basis for a vector space V, the vectors need to be linearly independent and they need to span V. You're showing the vectors are linearly independent, but you still need to show they span V.
 
  • #4
vela said:
For a set of vectors to be a basis for a vector space V, the vectors need to be linearly independent and they need to span V. You're showing the vectors are linearly independent, but you still need to show they span V.

My interpretation of the question was that the vectors needed to be a basis for some space, although not necessarily a basis for anything in particular. In that case, showing that they're linearly independent would be sufficient.
 
  • #5
ideasrule said:
My interpretation of the question was that the vectors needed to be a basis for some space, although not necessarily a basis for anything in particular. In that case, showing that they're linearly independent would be sufficient.

When exactly should I have to show that they span V as well as showing linear independency?
 
  • #6
If you want to prove the vectors form a basis, you always have to show they span the space. It's part of the definition of a basis.
 
  • #7
it's not enough to say a set is "a basis". you have to say what it is a basis FOR.

for example {1} is a basis for the space of all constant polynomials, but it is NOT a basis for the space of all polynomials of degree 1 or less (even though it IS a linearly independent set).

to elaborate, an mxn matrix A may, or may not, define a basis by its column vectors for the column space, and even so, the column space may not be the whole of the co-domain of A. some examples:

[1 0 0]
[0 1 0]. here, the columns are not linearly independent, but the first two form a basis for the column space, AND all of R^2.

[1 0]
[0 1]
[0 1]. here the columns are linearly independent, and form a basis for the column space, but NOT for all of R^3.
 
Last edited:
  • #8
ideasrule said:
My interpretation of the question was that the vectors needed to be a basis for some space, although not necessarily a basis for anything in particular. In that case, showing that they're linearly independent would be sufficient.
I would say that asking if a set of vectors is a basis without specifying which vector space it's supposed to be a basis of is a meaningless question. It's kind of like asking, is x=2 a solution? It doesn't really make sense to ask that without saying equation it's supposed to be a solution to.
 
  • #9
Sorry about not providing sufficient information. In my homework problems, I am asked to determine if a sequence of vectors form a basis of R^n.

By the definition of basis (linear independency and spanning sequence), I believe that it is impossible for any sequence of vectors to form a basis unless the sequence of vectors form a matrix that is n x n.

Is this correct?

Thanks.
 
  • #10
number0 said:
Sorry about not providing sufficient information. In my homework problems, I am asked to determine if a sequence of vectors form a basis of R^n.
Better terminology is a "set or collection" of vectors.
number0 said:
By the definition of basis (linear independency and spanning sequence), I believe that it is impossible for any sequence of vectors to form a basis unless the sequence of vectors form a matrix that is n x n.
The vectors don't have any direct bearing on matrices. The definition of a basis of a space (such as Rn) is that the vectors have to be a linearly independent set and they must span the space. For Rn, you need n vectors that are linearly independent.
number0 said:
Is this correct?

Thanks.
 
  • #11
Mark44 said:
Better terminology is a "set or collection" of vectors.
The vectors don't have any direct bearing on matrices. The definition of a basis of a space (such as Rn) is that the vectors have to be a linearly independent set and they must span the space. For Rn, you need n vectors that are linearly independent.

What happens if I have "too little" n-vectors? I guess that the set of vectors cannot be spanned in R^n.
What happens if I have "too much" n-vectors? I guess that the set of vectors is linearly dependent.

Is this the correct way to look at my homework problems?
 
  • #12
If you have too few vectors, the set of vectors can't possibly span whatever space we're talking about, and thus can't be a basis for the vector space. If the vectors happen to be linearly independent, they span a subspace of the vector space, and would be a basis for that subspace.

If you have too many vectors, the set of vectors can't possibly be linearly independent, and so aren't a basis for the vector space. They might span the vector space.
 
  • #13
there are 3 types of basis to my understanding, when i first learned this i was confused
1. basis of solution set
2. basis of null space/column space/row space
3. basis a span
 
  • #14
By "solution set," I take it you mean the solution set to a system of linear equations. If so, the solution set, null space, column space, row space, and span of a set of vectors are all vector spaces, so it's the same concept in each case: you're looking for a linearly independent collection of vectors that span the given space.
 
  • #15
Mark44 said:
If you have too few vectors, the set of vectors can't possibly span whatever space we're talking about, and thus can't be a basis for the vector space. If the vectors happen to be linearly independent, they span a subspace of the vector space, and would be a basis for that subspace.

If you have too many vectors, the set of vectors can't possibly be linearly independent, and so aren't a basis for the vector space. They might span the vector space.

Thank you. This helps so much.
 

1. What is a basis in linear algebra?

A basis in linear algebra is a set of linearly independent vectors that can be used to represent any vector in a vector space. These vectors form the building blocks for other vectors in the space.

2. How do you determine if a sequence of vectors is a basis?

To determine if a sequence of vectors is a basis, you can use the concept of linear independence. This means that all vectors in the sequence must be linearly independent, meaning that none of the vectors can be written as a linear combination of other vectors in the sequence. You can also check if the vectors span the entire vector space, meaning that any vector in the space can be expressed as a linear combination of the basis vectors. If both of these conditions are met, then the sequence of vectors is a basis.

3. What is the importance of a basis in linear algebra?

A basis is important in linear algebra because it provides a way to represent and manipulate vectors in a vector space. It also allows for the simplification of calculations and the solution of equations involving vectors. Additionally, a basis can help determine the dimension of a vector space.

4. Can a sequence of vectors be a basis for more than one vector space?

Yes, a sequence of vectors can be a basis for more than one vector space as long as the vectors meet the criteria for a basis in each space. This means that the vectors must be linearly independent and span the entire vector space in each case.

5. Is there a limit to the number of vectors in a basis?

The number of vectors in a basis can vary depending on the dimension of the vector space. For a vector space with n dimensions, a basis can have up to n vectors. However, it is also possible to have a basis with fewer vectors if the vectors are linearly independent and span the entire space.

Similar threads

  • Calculus and Beyond Homework Help
Replies
0
Views
418
  • Calculus and Beyond Homework Help
Replies
14
Views
532
  • Calculus and Beyond Homework Help
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
Replies
15
Views
898
  • Calculus and Beyond Homework Help
Replies
10
Views
2K
  • Calculus and Beyond Homework Help
Replies
15
Views
2K
  • Calculus and Beyond Homework Help
Replies
18
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
462
  • Calculus and Beyond Homework Help
Replies
5
Views
1K
Back
Top