People typically look at the concept of independence this way: “If I do something by myself, alone, then I’m independent; if I do something at all in collaboration with someone else, I’m not independent.”
This is erroneous. While being independent sometimes involves, as a byproduct, doing something on your own, this is not the definition or core of independence. To be independent means to think objectively, for yourself–and to be willing to accept the responsibility of judgment for matters important to your life, great and small.
You can be independent in a crowded room, or all by yourself. What makes you independent is your willingness to think.