Hello MHF, would appreciate some help with the following equestions;
Determine which of the following sets are subspaces. Give reasons for your answers.
(a)
(b)
Hello,
You have to check that for any u,v belonging to the set, u+v belongs to the set, and for any real number , belongs to the set.
So let and , where and .(a)
Does this belong to ?
Consider the case when
Let and(b)
Can the first coordinate be in the form ?
The problem here is that s and t are the same in the first and the second coordinates.
If there weren't, you could have just written
But you would have to get as your second coordinate...
A subset of a vector space is a subspace if and only if it is "closed" under vector addition and scalar multiplication.
If u and v are two members of H, they are of the form and for some positive numbers s and t. Their sum is while, for any number k, . What happens if k is negative?
If u and v are two members of L, they are of the form and where s, t, x, y can be any real numbers. Their sum is and, for a real number k, . Can you see that those are NOT of the correct form for L?