Recent content by cleaf

So you mean the U=[0] can only be represented by 0? By definition, T'([0]) = T(0), T'([a]) = T(a). If we always translate [a] to [0], it is correct. However, there is no rule against the direct sending, where T(a) != T(0) and ambiguity emerges. Please see the attached example 2.3.22 in the...

Wonderful exposition！ Thanks a lot! By definition, I can not understand why T(a) = T(b) if a and b are both in U. That is T'([0]) = T(0), T'([a∈U, a != 0]) = T(a), why T(0) - T(a) = T(0)? If we assume any element in U must be sent to zero in W, then U must be a subspace of Ker T.

I'm trying to prove the associative law of symmetric difference (AΔ(BΔc) = (AΔB)ΔC ) with other relations of sets. A naive way is to compare the truth table of two sides. However, I think the symmetric difference is not a basic one, it is constructed form other relations, that is AΔB =...

Thanks very much! I think the key point is T' is not a well defined map. As defined, for any equivalent class [a]=[c]={a, c, ...}, T'([a]) = T(a), T'([c]) = T(c), However, a-c is not guaranteed to be in Ker T (it is in U, but U is not Ker T). As a result, T'([a]) - T'([c]) = T(a-c) may not be...

I am reading his excellent book "Mathematical Physics Part 1, Second Edition", which has benefited me a lot in many ways. However, I have a doubt about the correctness of the theorem 2.3.23, which states that for any subspace U of V, the map T ' : V/U -> T(V) defined by T ' ([a]) = T|a> is...