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Let `vecv = <1,4,2>` and `vecw = <3,1,-1>` .Explain why the Span of `vecv`...
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High School Teacher
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.
Posted by degeneratecircle on March 14, 2013 at 10:34 PM (Answer #1)
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