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How do I change the spelling of my middle name in MN? - name change mn

My family and I have always said my name was Lyn. This is the way my mother wanted it in writing, that's how I grew up in writing, signing, etc.

When I finally took my birth certificate to obtain a passport, we learn that on my birth certificate that says that my real name is Lynn. This is how it appears under my Social Security number. Apparently he made a mistake when I was born in the hospital and not say what my parents did, and did not understand the error.

How can I just change the spelling of my name in Minnesota? I am a legal adult, but I hope to avoid expensive fees.