Span(S) is the intersection of all subspaces of V containing S

In summary, the theorem states that the span of a set S is the intersection of all subspaces of V containing S. This means that if W is any subspace of V that contains S, then the span of S is a subset of W. This also applies to any collection of subspaces of V containing S, regardless of the number of subspaces or what they are indexed by.
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JD_PM
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Homework Statement:: I want to understand the proof for the following theorem: span(S) is the intersection of all subspaces of V containing S.
Relevant Equations:: N/A

I know that if ##W## is any subspace of ##V## containing ##S## then ##\text{span}(S) \subseteq W##.

I have read (Page 157: # 4.86) that it follows that ##S## is contained in the intersection of all subspaces containing ##S## but I do not quite get why.

Once I understand the above I should be ready to move forward

Thanks! :biggrin:
 
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This is just a general statement about sets. If ##V\subset W_i## for any collection of sets ##W_i## (the index here doesn't have to be the natural numbers, it can be uncountable), then
$$V\subset \bigcap_i W_i.$$
 
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FAQ: Span(S) is the intersection of all subspaces of V containing S

What does it mean for a span to be the intersection of all subspaces of V containing S?

This means that the span of S is the smallest subspace of V that contains all the elements of S. In other words, it is the common set of all subspaces of V that contain S.

Why is the span of a set S important in linear algebra?

The span of a set S is important because it allows us to determine the dimension of the subspace generated by S. This helps us understand the structure and properties of the vector space V.

How do you find the span of a set S?

To find the span of a set S, we need to find all possible linear combinations of the elements in S. This means multiplying each element by a scalar and adding them together. The resulting set is the span of S.

Can the span of a set S be a subspace of V?

Yes, the span of a set S is always a subspace of V. This is because it contains all possible linear combinations of the elements in S, which satisfies the three properties of a subspace: closure under addition, closure under scalar multiplication, and containing the zero vector.

Is the span of a set S unique?

No, the span of a set S is not unique. This is because there can be multiple sets of elements that generate the same subspace. However, the dimension of the subspace generated by S is always unique.

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