How can a set of vectors that do not lie in a common plane span R^3, whereas a set of vectors that lie on a common plane do not span R^3

Question

anonymous · Answer

does it just have to do with the vectors that are in the set, and wheteher they can express a vectror in R^3 as a linear combination?

phi · Answer

if all the vectors line in the same plane, then you can only get to points on the plane (all linear combinations are still on the plane)
to get to a point off the plane you need  a vector that points up or down i.e. not in the plane

phi · Answer

so if you have a set of vectors in a plane  they span R^2  and one more vector not in the plane allows you to span R^3  (you can get to any point in R^3 if you scale your vectors the right way.)

anonymous · Answer

okay, i am starting to see how its. So if all three vectors lie in the same plane, then we can only get two points by way of linear combination, but R^3 has vectors of the form (v1,v2,v3), so then that set would not span R^3

phi · Answer

Your vectors in the plane could be 3-components  (in R^3) , but that does not allow you to get to all points in R^3  if the vectors all lie in the same plane.

phi · Answer

The easiest way to see this is imagine vectors of the form (x,y,0)
no matter how you combine them, you will never get a z value other than 0. You need a vector with a z ≠ 0 to get to all the points above or below the plane

anonymous · Answer

Right, so if all three vectors lie on that common plane,then we would only be taking into account the points in that specific plane (in which the three vectors lie). Wheareas if we have a set thaat has vectors that do not lie in the commone plane that allows us to get all the points in R^3

anonymous · Answer

meaing we can write any vector in R^3 as a linear combination of the vecorts in S(that do not lie on a common plane)

phi · Answer

yes

anonymous · Answer

I'm not trained in the lingo, but am imagining a sheet of glass (the plane) with the vectors on. They're not going to span R^3 without, as phi says, going up or down.

anonymous · Answer

good example, thanks you phi, and mcnalljj

phi · Answer

yw