Linear Algebra linear combinations help

[porschedude]

The linear combinations of v=(a,b) and w=(c,d) fill the plane unless _____.
Find four vectors u, v, w, z with four components each so that their combinations cu+dv+ew+fz produce all vectors (b₁, b₂, b₃, b₄) in four dimensional space.

I think that the first part of the answer, that fills the blank, is unless v is a scalar multiple of w, or vice versa.

But as far as the second part of the question, I have no idea what it is even asking. Are b₁, etc. referencing the first part of the question? Or is it just asking to write vectors that have all zero components except b₁ etc. Any help is much appreciated?

 

[Dick]

The first answer looks fine. I THINK the second one is just asking you to write down four vectors that span R^4. Like u=(1,0,0,0) might be a good choice for the first one. Can you give me a v, w and z that span together with u?

 

[porschedude]

That's what I thought it might mean v=(0,1,0,0), w=(0,0,1,0), z=(0,0,01), but that seems too simple. It's listed in the book as a challenge problems

 

[Dick]

That's what I thought it might mean v=(0,1,0,0), w=(0,0,1,0), z=(0,0,01), but that seems too simple. It's listed in the book as a challenge problems

Sure. But I can't think what else it might mean. It certainly can't be referencing any symbols in the first part, since the first part is about R^2.

 

[porschedude]

Alright, thanks for the help,