Proving Linear Independence and Span of D

In summary, the conversation discusses proving the existence of a subset D' of a nonempty subset D of a vector space V over a field F. This subset D' must have the same number of elements as a finite linearly independent subset B of span D. The goal is to show that the span of the set formed by taking the difference between D and D' and adding in B is equal to the span of D. The conversation also mentions that if D is linearly independent, then (D-D') U B is also linearly independent. One person shares that they have completed the proof, which is about 2 pages in length. They also mention that even the base case for 1 element had to be modified.
  • #1
andytoh
359
3
I can't seem to figure this one out:

Question: Let D be a nonempty subset of a vector space V over a field F. Let B be a finite linearly independet subset of span D having n elements. Prove there exists a subset D' of D also having n elements such that

span[(D-D') U B] = span(D).

Moreover, if D is linearly independent, so is (D-D') U B.

Can anyone help?
 
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  • #2
Here's the beginning of my induction proof for the case of 1 and 2 elements. I'm working on the nth step now.
 

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  • #3
Ok, I've finished the proof now.

It is about 2 pages long! Even the base case for 1 element had to be modified to some length.
 
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1. How do you prove linear independence of a set of vectors in D?

To prove linear independence of a set of vectors in D, you can use the definition of linear independence, which states that a set of vectors is linearly independent if the only solution to the equation c1v1 + c2v2 + ... + cnvn = 0 is c1 = c2 = ... = cn = 0, where v1, v2, ..., vn are the vectors in the set and c1, c2, ..., cn are scalars. This means that none of the vectors in the set can be written as a linear combination of the others.

2. What is the difference between linear independence and linear dependence?

The difference between linear independence and linear dependence lies in the number of solutions to the equation c1v1 + c2v2 + ... + cnvn = 0. In a linearly independent set of vectors, the only solution is when all the coefficients are 0. In a linearly dependent set, there are infinitely many solutions, meaning that at least one vector in the set can be written as a linear combination of the others.

3. How do you prove the span of a set of vectors in D?

To prove the span of a set of vectors in D, you can show that every vector in D can be written as a linear combination of the vectors in the set. This means that any vector in D can be reached by taking a linear combination of the set's vectors, which shows that the set spans D.

4. Can a set of vectors be linearly independent and span D at the same time?

Yes, a set of vectors can be both linearly independent and span D at the same time. This means that the set of vectors is a basis for D, which is a set of linearly independent vectors that span the entire vector space.

5. How does proving linear independence and span of D relate to solving systems of linear equations?

Proving linear independence and span of D can be useful in solving systems of linear equations because it allows us to determine if a set of vectors is a basis for D. If the set of vectors is a basis, then it can be used to construct a unique solution for any system of linear equations in D. Additionally, the concept of linear independence is closely related to the concept of linearly independent equations, which is used in solving systems of equations using methods such as Gauss-Jordan elimination or Cramer's rule.

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