Spanning set & Linearly (In)dependent Question below. I'm stuck. I don't know what's next...

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Spanning set & Linearly (In)dependent Question below. I'm stuck. I don't know what's next...

Mathematics
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2 vectors define a plane, a 2d object ... can a 2d object touch all of 3d space?
and independant set is one in which each vector cannot be defined in terms of other vectors in the set

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no? 2d can't span all 3d space, so it is not a spanning set of R^3? so since c2 needs c1, then it is linearly dependent?
since 2,4,0 is not a scaled version of 2,1,3 the vectors are independant.
one way to consider it is ... a v1 + b v2 = v3 for all vectors (v3) in R^3 in order to span (to cross over, to reach all points) of R^3, you have to be able to create them from the given conditions ...
a(2,1,3) + b(2,4,0) = (x,y,z) 2a + 2b = x a + 4b = y 3a + 0b = z can the vector (0,1,3) be obtained? 2a + 2b = 0; b=-1 a + 4b = 1; 1-4 doest eqaul 1 3a + 0b = 3 ; a = 1
think of a piece of paper, you can draw 2 arrows on it that can be combined to reach all points in that paper ... but in order to get off of the paper you need to add a vector that is not contained on the paper ... a vector that points above/below it.
oh okay... i think i understand it better. two vectors = only spans a plane so if in case i'm given three vectors, there will be a possibility that they can span R^3 right?
it requires n, independant vectors, to span R^n ... yes
... at least might need to be included for rigor
two independant vectors create a plane yes
for the linearly independent/dependent, |dw:1443917976763:dw|
oh i think it makes sense to me now
correct enough yes
okay thank you again! :)
youre welcome

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