In each of the following, carefully justify whether the set H is a subpace of vector space V. If H is infact a subspace, give a basis for H.

1.) V = R^4, H = {(a,b,c,d) "is an element of" R^4 | a + b + c + d = 0}

2.) V = R^3, H = {(a,b,c) "is an element of" R^3 | abc = 0}

I think there are 3 properties to check? Seeing if 0 "is an element of" H; Seeing if u + v "is an element of" H, and if u "is an element of H" then is cu "an element of H" for c "an element of" R.