I have to use the theorem which states:
If W is a set of one or more vectors in a vector space V, then W is a subspace of V if and only if the following conditions hold.
a) If u and v are vectors in W, then u + v is in W.
b) If k is any scalar and u is any vector in W then k*u* is in W.
to find if all vectors of the form $(a, b, c)$, where $b = a + c$, subspaces of $R^3$.
I understand how to do it for the first two problems which were of the form $(a, 0, 0)$ and $(a, 1, 0)$, but don't understand for this form.
For example for $(a, 1, 0) + (d, 1, 0) = (a + d, 2, 0)$, which is not in the correct form.
But for $(a, b, c)$ I am not sure what to make of it.
I get $(a, b, c) + (d, e, f) = (a + d, b + e, c + f)$
$(b+e) = (a+c) + (d+f)$, or $(a+d) + (c+f)$
I eventually get $(a, b, c) + (d, e, f) = (a+d, [(a+d) + (c+f)], c+f)$
I see a pattern there, but I no longer recognize the form and don't understand what it's telling me.
The answer key in the book says that it is not a subspace of $R^3$.
How should I be using Part A of the theorem?