I'm self-studying abstract algebra, and prior to fields there's a brief section on vector spaces. One of the questions asks:
"Is $U = \{(a, b-1, c)| a, b, c \in F \}$ a subspace of $F^3$? ($F$ a field)"
I figured that the answer was yes because $(a, b-1, c) + (d, e-1, f) = ((a+d), (b+e-1) -1, (c+f))$ and $k(a, b-1, c) = (ka, (kb -k +1) - 1, kc)$
so the set would be closed under addition and scalar multiplication. Why, then, is this answer wrong? It's apparently not a subspace? I feel like I must have some crucial misunderstanding about this topic, and I don't have anyone else to help me. What's the gap in my understanding?