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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[,,]+b[,,[-1]],` `vecy=c[,,]+d[,,[-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[,,]+n[,,[-1]]` for some values of `m` and `n.` The verification of this is pretty straightforward so I'll leave out the details.
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