I had a course in linear algebra in which the concept of direct sum was presented. At first, it does not seems too complicated, there were some exercises to identify if things are or aren't direct sums but there was nothing beyond that. So, why are they important?
I know that summing a vector from a vector space and a vector from another vector space in which these vector spaces share only the $0$ vector will give me a new vector space but I guess I could create this notion (creating new vector spaces) only by talking of linear combinations.