
Determining Subspace
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.

That's right, H has to be nonempty (easy to check: 0 has to be an element) and you can combine the other two properties into one: it has to be closed under lineair combinations, i.e. if v and w (vectors) are solutions, then kv+lw (k,l scalars) has to be a solution too.
1) 0 is a solution, trivial. If (a,b,c,d) and (e,f,g,h) are solutions, is k(a,b,c,d)+l(e,f,h,h) a solution too? Check it :)
2) Try it.

Hmm. Okay so there are 3 things you have to check for each. So I believe its something along check if the zero vector is an element of H, and it H is "closed" under adddition and multiplication.
I dont really get what the a + b + c + d = 0 is for... it seems obvious that H is a subspace for a...what is the basis for it? How do I show it?
#2: again, like #1, it seems obvious H is a subspace...same questions. A detailed answer will help me do some of the other homework problems.
Thanks

Tried doing some other of the others, though they're all like this and ask the same question. I thought it would be sufficient to prove if the 0 zero is an element of H only, but not according to my prof...
Anyone familiar with subspace?

Taking the two vectors I suggested, we want to know whether the following is in H too:
$\displaystyle k\left( {a,b,c,d} \right) + l\left( {e,f,g,h} \right) = \left( {ka + le,kb + lf,kc + \lg ,kd + lh} \right)$
This is only true if ka+le + kb+lf + kc+lg + kd+lh = 0. But we can rewrite:
$\displaystyle ka + le + kb + lf + kc + \lg + kd + lh = k\underbrace {\left( {a + b + c + d} \right)}_{0} + l\underbrace {\left( {e + f + g + h} \right)}_{0} = 0$
Which is exactly what we wanted, so H is indeed a subspace.