Proving Uniqueness in Subspace Addition

[TranscendArcu]

Homework Statement

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The Attempt at a Solution

So we have that A + B is a vector in S + T, where A is an element of S and B is an element of T. Suppose there is another vector A' + B' also in S + T, where A' is an element of S and B' is an element of T. Let A + B = A' + B' to suppose that the sum cannot be written uniquely. This implies that A + B - A' - B' = 0. This implies that A - A' + B - B' = 0. This implies that B - B' is the additive inverse of A - A', but this is only true if A - A' and B - B' are both in the same subspace. Therefore, A - A',B - B' must both be elements of S,T. But, by definition, the only element in both S,T is 0. Therefore, the only vector that can be written as A + B = A' + B' is the zero vector, which is necessarily unique as a consequence of the properties of subspaces. Therefore, all vectors can be written as unique combinations of A + B.

Am I doing this right? I haven't done the second part yet.

 

[mighty2000]

Yes, you are on the right track! Your explanation for why A-A' and B-B' must both be in S and T is correct. For the second part, you can show that A + B = A' + B' implies A = A' and B = B', which would also show uniqueness. Keep up the good work!