Question

hello everyone :) let V be a vector space prove that for every W (subset of V) there is a linearly independent S (subset of W) such that span(W)=span(S) ?

Answer 1

Do you understand what the question is asking?

Answer 2

as i recall, take everything in S and start throwing out stuff

Answer 3

If you take a random group of vectors in this vector space, say:\[W=(v_1,v_2,\ldots,v_k)\] These vector may or may not be linearly independent. If I take the span though:\[span(v_1,v_2,\ldots, v_k)\]that should equal the span of a linearly independent set. you need to show that.

Answer 4

given that span(W)=span(S) and it's required to prove that S is a linearly independence set

Answer 5

Lets say I gave you the three vectors, (1,2,3), (4,5,6), and (5, 7, 9). Note these are linearly dependent since the first two sum to give the third. How would you show that these are linearly dependent?

Answer 6

let v1=(1,2,3) , v2= (4,5,6) and v3=(5, 7, 9) then I have v3=v1+v2 i.e. v3 is a linear combination of v1 , v2 so {v1,v2,v3} is a linearly dependent set of vectors

Answer 7

Right. So when we look at the span of the three vectors, it turns out we can throw away the third, since it is giving redundant information. So just take the span of the first two, and that is a linearly independent set.

Answer 8

Basically for your proof, you want to outline that process. Take an arbitrary set of vectors W. Find out which are actually linearly independent, and which are dependent. Throw away the dependent vectors. Make this new set of vectors S. Now, given a set W, you have created a set S where S is filled with linearly independent vectors, and Span(W) = Span(S).

Answer 9

Just make it sound fancier in the proof lol.

Answer 10

ok i will try