Understanding Subspace Basis and Counterexample

In summary, the conversation discusses the claim made in a book about vector spaces, stating that if a basis for the vector space \mathbb{R}^4 is given and a subspace W is defined, there exists a W whose basis is not a subset of the given basis for \mathbb{R}^4. The book provides a proof by counterexample and clarifies that the vectors in the given basis are not necessarily in W. It also mentions that there can be multiple bases for a vector space and that a basis for W can be extended to a basis for the entire space, but there can also be bases that do not contain any vectors from the basis for W.
  • #1
Caspian
15
0
My book made the following claim... but I don't understand why it's true:

If [tex]v_1, v_2, v_3, v_4[/tex] is a basis for the vector space [tex]\mathbb{R}^4[/tex], and if [tex]W[/tex] is a subspace, then there exists a [tex]W[/tex] which has a basis which is not some subset of the [tex]v[/tex]'s.

The book provided a proof by counterexample: Let [tex]v_1 = (1, 0, 0, 0) ... v_2 = (0, 0, 0, 1)[/tex]. If [tex]W[/tex] is the line through [tex](1, 2, 3, 4)[/tex], then none of the [tex]v[/tex]'s are in [tex]W[/tex].

Is it just me, or does this not make any sense? First of all, (1,2,3,4) is a linear combination of (1,0,0,0)...(0,0,0,1), isn't it?

I'm very confused...

Any help would be greatly appreciated :).
 
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  • #2
Yes, but that's not what they're saying. They're saying that none of the four [itex]v_i[/itex] vectors are in W. They're also saying that a basis vector of W (which must be a multiple of (1,2,3,4)) can't be equal to one of the [itex]v_i[/itex].

When they talk about the set of "v's" they really mean a set that only has four members, not the subspace they span.
 
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  • #3
Ok, ok... I get it. (1,2,3,4) is in the vector space spanned by (1,0,0,0)...(0,0,0,1), but these vectors aren't a basis for the subspace which is the line through (1,2,3,4) because these vectors span too much space.

Ok, I guess this was a dumb question. Thanks for your help :).
 
  • #4
I don't think you fully get it yet.
Caspian said:
Ok, ok... I get it. (1,2,3,4) is in the vector space spanned by (1,0,0,0)...(0,0,0,1), but these vectors aren't a basis for the subspace which is the line through (1,2,3,4) because these vectors span too much space.
It should be obvious that B={(1,0,0,0),...,(0,0,0,1)} isn't a basis for W, since B spans the whole space R^4! So they can't be a basis for any proper subspace of V (indeed, they span "too much space").

But that's not what your book is asserting. They are only talking about a subset of B={(1,0,0,0),...,(0,0,0,1)}. So they're saying that even some subset of B cannot be a basis of W. Remember that a basis of W first of all consists of elements of W. None of the vectors in B are in W.
 
  • #5
The point you need to keep in mind is that there exist an infinite number of different bases for any given vector space. If W is a subspace of V and we are given a basis for W, then we can extend that to a basis for V. That is, the basis for V will consist of all vectors in the basis for W together with some other vectors. But there can also exist bases for V that do not contain any of the vectors in the basis for W.
 

What is subspace basis?

Subspace basis is a set of vectors that can be used to span a subspace, or a smaller vector space within a larger vector space. These vectors are linearly independent, meaning that none of them can be written as a linear combination of the others.

How is subspace basis determined?

Subspace basis can be determined by finding a set of linearly independent vectors that span the subspace. This can be done by creating a matrix from the vectors and reducing it to its row echelon form. The non-zero rows of the reduced matrix will form the basis.

What is a counterexample in subspace basis?

A counterexample in subspace basis is a set of vectors that are not linearly independent but still span the subspace. This means that they are not a valid basis for the subspace, even though they satisfy the criteria of spanning it.

What is the importance of understanding subspace basis?

Understanding subspace basis is important for solving problems in linear algebra, as it allows us to work with smaller vector spaces within a larger space. It also helps us to understand the concept of linear independence and how it relates to vector spaces.

How can understanding subspace basis be applied in real-world scenarios?

Subspace basis can be applied in various fields such as physics, engineering, and computer science. In physics, it is used to understand the concept of forces in different directions. In engineering, it is used to analyze structural stability and design mechanisms. In computer science, it is used to process large amounts of data efficiently and to create algorithms for machine learning.

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