Is W a Subspace of R^3 and What is Its Geometry?

[justa18unlv]

I'm stuck on a problem which asks:
Determine whether W is a subspace of R^3. If W is a subspace, then give geometric description of W. The problem is W={x:x3=2x1-x2} and x=[x1, x2, x3]
I tried solving it but I'm having a hard time understanding the properties of R^n and using them. I guess I'm not as good with proof as I'm with numbers. Please help.

 

[HallsofIvy]

Because x3= 2x1- x2, any vector in that subspace can be written as [x1, x2, x3]= [x1, x2, 2x1- x2]= [x1, 0, 2x1]+ [0, x2, -x2]= x1[1, 0, 2]+ x2[0, 1, -1].

Can you see what a basis for that subspace is? Of course, 2x1- x2- x3= 0 is the equation of a plane.