Howdy everyone. I'm not very good at writing proofs, so I am wondering if someone can tell me if I'm even on the right page with this. I am not sure if I understand the idea correctly. The theorem goes as follows: Suppose V is a finite dim. vector space with subspaces U and W such that V is the direct sum of U and W. Let Z also be a subspace of V. Then Z cannot equal the direct sums of (i) the intersection of Z and U and (ii) the intersection of Z and W unless Z is a subspace of U or W.(adsbygoogle = window.adsbygoogle || []).push({});

Here's my thoughts:

Let "C" denote the direct sum of the intersections (i) and (ii) above. Then two conditions must hold:

1) The intersection of the intersections (i) and (ii) must be the zero vector; and

2) The sum of the intersections must equal V.

The elements of C are given as {x|x is in Z and x is in U and x is in W}={0v}. Thus every x in Z that intersects U and W is in C. But we have that C={0v} since it is a direct sum. The only way that {0v}=(ZintersectU)intersect(ZintersectW) is if Z contains only the zero vector. Thus Z is the zero subspace, and it follows that Z must be a subspace of U and W.

Yeah, let me know if that's right... I'm not sure that I am correct in asserting that Z contains only the zero vector.

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# Correct reasoning about direct sums proof?

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