# Vector Spaces

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 Preview

(1) Is true. Since every element of W1 is orthogonal to every element of W2 is suffices to ...

#### 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.