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.

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23 July, 02:27

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.

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Amaris Salinas · Answer 1

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.