Interesting Linear Algebra Problem

[Newtime]

Homework Statement

Let x=(1,0,0) y=(2,1,0) and z=(2,2,1) be column vectors in R³. Consider the set V={tx+sy+uz 0\leqt,s,u\leq1}. What does this look like specifically?

Homework Equations

n/a

The Attempt at a Solution

No work here, just a thinking problem, but I thought that the set V would fill the solid (a paralellapipid?) formed by the three vectors x,y, and z. Is this correct? Other than visualization, I suppose a reason for this would be that it is analagous to a similar problem in R² in which the set vills the parallelogram formed by the two vectors. Thanks.

 

[Mark44]

If the given vectors were linearly independent, set V would be a three-D region in space, but if they are linearly dependent V will be only a one- or two-dimensional subset of R³.

 

[Newtime]

If the given vectors were linearly independent, set V would be a three-D region in space, but if they are linearly dependent V will be only a one- or two-dimensional subset of R³.

They are linearly independent. And I know they fill some space in R³ but the question was which space...as into describe it. So my best explanation is the vectors fill the paralellepiped formed by the three vectors. Isn't this correct?

 

[Dick]

They are linearly independent. And I know they fill some space in R³ but the question was which space...as into describe it. So my best explanation is the vectors fill the paralellepiped formed by the three vectors. Isn't this correct?

Yes, you are visualizing it correctly.

 

[Newtime]

Yes, you are visualizing it correctly.

Thanks, I figured I was but wanted to check with all the gurus here.