How do I tell, geometrically, that these vectors span a line, plane, or 3D space?

Vectors (1,-1,0), (0,1,-1), (1,0,-1)
Vectors (1,-1,0), (0,1,-1), (1,0,1)

We haven't got to row reduction / linear independence stuff yet.

Question

How do I tell, geometrically, that these vectors span a line, plane, or 3D space?

Vectors (1,-1,0), (0,1,-1), (1,0,-1)
Vectors (1,-1,0), (0,1,-1), (1,0,1)

We haven't got to row reduction / linear independence stuff yet.

zzr0ck3r · Answer

are they lin ind?

zzr0ck3r · Answer

put them in a matrix and row reduce

zzr0ck3r · Answer

wow just reading the rest....

mrrobot3 · Answer

So if I put them in RREF, how do I tell that they span a plane or a 3d space?

zzr0ck3r · Answer

what did you get?

zzr0ck3r · Answer

sec phone...

mrrobot3 · Answer

Vectors (1,0,0) (0,1,0) (1,1,0)

Vectors (1,0,0) (0,1,0) (0,0,1)

mrrobot3 · Answer

^-- Those are two different matrices.

zzr0ck3r · Answer

so the first matrix has 2 linearly independent vectors because the 3rd one is the sum of the first two, so this will span a plane

the second matrix has 3 lin ind vectors (none can be made by adding the other 2 along with scalar multiplication). So this will span 3 space.