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

[number0]

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.

 

[ideasrule]

Yes, because by definition every element of a basis is linearly independent of the others.

 

[vela]

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.

 

[ideasrule]

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.

 

[number0]

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?

 

[vela]

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.

 

[Deveno]

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.

 

[vela]

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.

 

[number0]

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.

 

[Mark44]

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.

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.

Is this correct?

Thanks.

 

[number0]

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?

 

[Mark44]

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.

 

[cloud360]

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

 

[vela]

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.

 

[number0]

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.