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
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! :)