Prove that if W is a (non zero) subspace of Rn with basis B = {w1...wk} then W p

Question

Prove that if W is a (non zero) subspace of Rn with basis B = {w1...wk} then W perp the orthogonal complement of W is equivalent to: {v e Rn | v dot w = 0 for all w e W}  (e means exist) (Hint: Start with the actual def of W perp and show that these 2 sets contain the same elements)

Attempt:

if W is the span(S) with S = ({w1...wk}), and that B = {w1....k} as the basis of W, then W perp = {v e Rn | v dot w = 0 for all w e W} by def in the book. So w perp contains vectors v1...vk so that

w1 dot v1 = 0

...

wk dot vk = 0

(and that's it... I'm not even sure how to do this prove to be honest....I figured that the question is asking for the prove of a definition, how can we even prove the definition of w perp... isn't it given? The hint is also throwing me off, what are the 2 sets that contains the same elements do I need to compare with?... maybe I'm mistaken)

Explanation / Answer

I think the question is little different
May, its asking that   {v e Rn | v dot w = 0 for all w e B} is orthogonal complement of W

iff {v e Rn | v dot w = 0 for all w e W} is the orthogonal complement of W

Can you please check the question in book once more. Something might be missing

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