Subspaces

[Imum Coeli]

I am not seeing what should be simple...

Question:
Which of the following sets are subspaces of R^3?

U1 = {t(1,2,3)|t belongs to R}
U2 = {(0,0,0),(1,0,0),(0,1,0),(0,0,1)}
U3 = {x,y,2x+4y)|x,y belong to R}
U4 = {(x,y,z) belongs to R^3|x+y-z=0, 2x+y+z=0}
U5 = {(x,y,z) belongs to R^3|x+y-z=1}
U6 = {(x,y,xy)|x,y belong to R}

Notes:
First, a subspace must contain the zero vector and be closed under addition and scalar multiplication.

Now my (incorrect) attempt at an answer.

U1 is a subspace because it is a line through the origin.
U2 is a subspace because it is a plane through through the origin
U3 is a subspace because it is all of R^3
U4 is a subspace because it is the solution set for two homogeneous linear equations
U5 is not a subspace because it does not contain the zero vector.
U6 is a subspace because it is all of R^3

I believe I am missing something obvious...
Thanks for any advice.

 

[daon2]

"Line through the origin" is only an acceptable answer if you show that a line through the origin IS a subspace. Show it contains zero, and is closed under sums and scalar products.

U2 is not a plane through the origin. It is only four vectors.

U3 is not all of R^3. It is spanned by the vectors (1,0,2) and (0,1,4).

U6 is again not all of R^3, since it does not contain (0,0,1). That still does not imply it is or is not a subspace.

Attempt to show all three properties for each. If it fails any one of them, it is not a subspace.

 

[Imum Coeli]

U1 and U3 are subspaces (I can show this).

I think that U6 is not a subspace because it is not closed under addition.
i.e., W = {(x,y,xy)|x,y belong to R}
Let u = (1,2,2) belong to set W and let v = (2,4,8) belong to W
then u + v = (3,6,10) which is not in W.

U2 is not a subspace but I don't know where to start.

Thanks for your help.

 

[daon2]

Your explanation of U6 is fine. It might have been a bit easier, just multiplying (1,1,1) by 2.

U2 is probably the easiest one on the list. Pick any nonzero vector in the set and multiply by 2. Is it still in the set? One thing you should note (but probably can't use as justification on you assignment) is that all non-zero subspaces of an R^n must be infinite, since if v is any nonzero vector in W then so is 2v, 3v, 4v, ... etc.

 

[Imum Coeli]

Thanks for your help.

(It's not an assignment, I'm just trying to get a better understanding of what's going on, so any and all of your advice is much appreciated.)