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1. I must verify the 0 vector is in S+T. Since S and T are subspaces, the 0 vector must exist in both S and T. Thus 0+0=0 and 0 vector is in S+T

2. I must verify that if X and Y are subspaces in S+T, then I need to check if X+Y is still in the subspace S+T.

Note: X and Y are of the form S+T, which yields the following:

So I define X as U1+U2, where U1 is in S and U2 is in T.

I define Y as V1+V2, where V1 is in S and V2 is in T.

X+Y=[U1+V1, U2+V2] Since U1 and V1 are both in S, then U1+V1 is in S. Since U2 and V2 are both in T, then U2+V2 are both in T. Thus X+Y is in S.

3. Lastly I must verify that if Y is in S+T, then I need to check if cY is in S+T, where c is some scalar. Using the same Y=V1+V2 where V1 is in S and V2 is in T. I get cY=c(V1+V2)=cV1+cV2, this satisfies that cV1 is in S and cV2 is in T. Thus cY is in S+T.

Is this correct?

Also, if a certain part is incorrect, can you refer clearly to the part or quote just that part? I get confused easily.

Thanks.