Proof of a theorem about spanning sets

In summary, the statement discusses the relationship between a subspace W of R^n and a subset A of R^n. It states that A spans W if and only if the set of all vectors in R^n that are dependent on A is equal to W. The book then proceeds to prove this by showing that if A spans W, then the set of all vectors dependent on A is equal to W. This is because every vector in W is dependent on A, which is defined as being in <A>. The conversation ends with a clarification on the meaning of "dependent on A."
  • #1
jwqwerty
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Homework Statement


Let W be a subspace of R^n and let A be a subset of R^n. Then A spans W if and only if <A> = W (<A> is the set of all vectors in R^n that are dependent on A). Prove it.

Homework Equations





The Attempt at a Solution


Ok the book goes likes this;
First, it proves that if A spans W, then <A>=W
It says that every vector in W is dependent on A, so W⊆<A>. But i do not get this part. Why is this true?
 
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  • #2
jwqwerty said:
<A> is the set of all vectors in R^n that are dependent on A
It says that every vector in W is dependent on A, so W⊆<A>. But i do not get this part. Why is this true?

Do you see it now?
 
  • #3
What does "dependent on A" mean?
 

FAQ: Proof of a theorem about spanning sets

1. What is a spanning set?

A spanning set is a set of vectors that can be used to represent all other vectors in a vector space. In other words, any vector in the vector space can be written as a linear combination of the vectors in the spanning set.

2. What is the purpose of proving a theorem about spanning sets?

The purpose of proving a theorem about spanning sets is to provide a mathematical proof that a given set of vectors is indeed a spanning set for a particular vector space. This helps to establish the properties and characteristics of the vector space and provides a way to systematically solve problems involving vector spaces.

3. How do you prove a theorem about spanning sets?

To prove a theorem about spanning sets, you must first clearly state the theorem and its assumptions. Then, you must use mathematical techniques such as algebra, linear transformations, and vector operations to show that the given set of vectors satisfies the conditions of the theorem and is therefore a spanning set for the vector space.

4. Why is it important to understand the properties of spanning sets?

Understanding the properties of spanning sets is important because it allows for a deeper understanding of vector spaces and their applications. It also helps in solving problems involving vector spaces and provides a foundation for further mathematical concepts such as linear independence and basis vectors.

5. Can there be more than one spanning set for a vector space?

Yes, there can be multiple spanning sets for a vector space. This is because a vector space can have an infinite number of bases, which are essentially spanning sets that are linearly independent. However, any spanning set for a vector space will have the same number of vectors as its dimension.

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