This problem is for Linear Algebra Done Right 3rd Edition:
Problem 18/Chapter 1:
Does the operation of addition on the subspaces of V have an additive identity? Which subspaces have additive inverses?
My doubt is in my attempt of answer for the first question:
Clearly for a Subspace $U$ of a vector space $V$ we have that the sum with the subspace $\{0\}$ implies:
$$U+\{0\}=U$$ (which is the answer given in previous posts)
However, problem 15 in the same chapter gives the result that $U+U=U$
We know that the additive identity is unique, then, Do the later results imply a contradiction? Since the addition of $U$ or $\{0\}$ gave the same result(both act as additive identity) for $U$.
I would appreciate if someone can illuminate this basic question
Best Answer
If $U\ne \{0\}$ then it's not an additive identity. It would have to satisfy $W + U = W$ for all subspaces $W$, not only for the case $W=U$.
There's no such thing as "identity for U".