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Amiraneli [1.4K]
2 years ago
6

Question 10. Suppose U and W are subspaces of V, dim V = n, dim U + dim W = n, and U NW = {0}. Prove that V=UW.

Mathematics
1 answer:
Bess [88]2 years ago
4 0

Answer: If the dimension of V is n, then V has n elements.

Now, dim(U) + dim(W) = n, this means that the addition of the dimensions of U and W also has n elements.

and because U and W are subspaces of V, you know that every element on U and W is also an element of V.

If U ∩ W = ∅, means that there are no elements in common between U and W.

Because there are no elements in common, then Dim(U) + Dim(W) = dim(U ∪ W) = n

So U ∪ W has the same number of elements as V, and every element of W and U is also an element of V

this means that U ∪ W = V.

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