Linear Algebra - dimension of orthogonal complement

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The discussion centers on a linear algebra problem regarding the dimension of an orthogonal complement. The user believes their solution is correct, concluding with answer C, while the answer key indicates D. There is a call for verification of the user's solution, suggesting a potential error in the answer key. The user expresses frustration over the discrepancy between their findings and the provided answer. Overall, the conversation highlights a conflict between a personal solution and an authoritative source.
GuiltySparks
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I've attached a copy of the problem and my attempt at a solution.

This seems like a relatively straightforward question to me, but my answer seems to be the exact opposite of what the answer key says.

I reach the conclusion that the answer is C, but the answer is apparently D.

I'm beginning to think that the answer key is wrong. Can anybody verify my solution?
 

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I think your answer key is wrong.
 

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