Recent content by ZenchiT

Thank you very much!

Thank you very much! You have cleared many misconceptions! It's always the really abstract type things that get me!

So if this is the case and b1 is linearly independent from a1,a2 then that would mean that AUB is not a subspace because it is not closed under addition? Is that correct? Thank you

Thank you so much! One last question! How would you go about proving linear dependence between the elements of different sets? For example if we take a1 = (3,3,0) and a2 = (2,2,0) and b1 (0,0,1) Then would we have, (3x1,3x1,0) + (2x2,2x2,0) = (0,0,1) So you would have the following systems of...

I see! I think that makes sense. What if we consider a case where we have 3 vectors a1,a2,b1 and we have A = span (a1 +a2) and B = span (b1) Then because neither of them are a basis would the same hold? For this example if b1 was contained within a1 and a2, i.e. x1a1 + x2a2 = b1, only then...

Thank you for your help again! Yes I understand that A and B are subspaces, however the difficulty that I'm finding, is where does this link with linear independence? If we can claim that AUB is a vector space if (a1,a2,a3) are linear independent then does that mean that for any element, a in A...

I know that it is a subset but is it a subspace? I.e, would this be closed under addition and scalar multiplication? My intuition says that wouldn't be the case because to be closed under addition a+b must be in AUB, however its not always the case that an element from span A and an element of...

Would that also imply that it is a subspace of R^3?

Ahh okay that makes sense! Would that mean that A U B = A + B?

How would the vectors being linearly independent affect anything?

Oh, I see so does that mean that they're are infinitely many elements? However don't all the elements follow two standard patterns? You'll have elements from span (A) = x1a1+x2a2+x3a3 and elements of span (B) y1b1+y2b2 where x1,x2,x3,y1,y2 are elements of the field?

Yes, we are given that they are in Euclidean space R^3. However when we do the union of the two things, surely we just have the two elements, the spans?

Sorry! Oh ok, so does A and B have infinitely many elements?

I know there is a theorem which states that a span is a subspace therefore addition and scalar multiplication are defined. Therefore is the number of elements 2?

They live inside the span? A span is a set of all linear combinations of all the elements inside it?