Let `vecv = <1,4,2>` and `vecw = <3,1,-1>` .Explain why the Span of `vecv` and `vecw` is a plane.

Expert Answers
degeneratecircle eNotes educator| Certified Educator

First of all, the span is two dimensional because the two vectors are linearly independent. This is especially easy to verify with only two vectors because neither is a multiple of the other.

As for it being a plane, i.e. "flat", I guess that depends on what your definition for that is. One way is to verify that if you take any two points `vecx,vecy` in the span, the line segment between them is entirely contained in the span.

So let `vec x=a[[1],[4],[2]]+b[[3],[1],[-1]],` `vecy=c[[1],[4],[2]]+d[[3],[1],[-1]]` .

The line segment between them contains all points

`vecx+t(vecy-vecx)` for `0<=t<=1.` Then using our above expressions for `vecx` and `vecy,` we just note that

`vecx+t(vecy-vecx)=m[[1],[4],[2]]+n[[3],[1],[-1]]` for some values of `m` and `n.` The verification of this is pretty straightforward so I'll leave out the details.