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

　
To be a subspace, set has to satisfy the following conditions:
1) It must contain the 0 vector
2) If u and v are elements of set, then u+v must be an element of set
3) If u is element of set, and k is a real number, then ku must be element of set

Note: if set satisfies conditions 2) and 3), then it also satisfies condition 1)
Proof: If u element of set, then so is −1*u = −u (by condition 3), and therefore, u+(−u) = u−u = 0 is also an element of set (by condition 2)

(U1) Yes. Closed under addition and scalar multiplication.

(U2) No. Not closed under either addition or scalar multiplication.
　　　　(1,0,0)+(0,1,0)=(1,1,0) ∉ U2; 2(1,0,0)=(2,0,0) ∉ U2

(U3) Yes. Closed under addition and scalar multiplication.
　　　　(a, b, 2a+4b) + (x, y, 2x+4y) = (a+x, b+y, 2(a+x)+4(b+y)) ∈ U3
　　　　k (x, y, 2x+4y) = (kx, ky, 2kx+4ky) ∈ U3

(U4) Yes. Closed under addition and scalar multiplication.

(U5) No. Not closed under either addition or scalar multiplication.
　　　　(a,b,c)+(x,y,z) = (a+x,b+y,c+z) ∉ U5, because:
　　　　　　(a+x)+(b+y)-(c+z) = (a+b-c)+(x+y-z) = 1+1 = 2
　　　　k(x,y,z) = (kx,ky,kz) ∉ U5, because kx+ky-kz=k(x+y-z)=k(1) = k

(U6) No. Not closed under either addition or scalar multiplication.


See if you can do the ones I didn't do.