Vector Spaces
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How to prove or counter with example the following statements:
(1) If two subspaces are orthogonal, then they are independent.
(2) If two subspaces are independent, then they are orthogonal.
I know that a vector v element of V is orthogonal to a subspace W element V if v is orthogonal to every w element W. Two subspaces W1 and W2 are said to be orthogonal subspaces if for every w1 element W1 and w2 element W2 the inner product satisfies (w1, w2)=0
I would appreciate if you could provide proving explanation on two of above statements.
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Solution Summary
This solution is comprised of a detailed explanation to prove or counter with example the following statements:
(1) If two subspaces are orthogonal, then they are independent.
(2) If two subspaces are independent, then they are orthogonal.
Solution Preview
(1) Is true. Since every element of W1 is orthogonal to every element of W2 is suffices to ...
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