All my life, I've learned to treat vectors as a set of 3 real numbers, which I multiply by a basis to get the actual vector: $\langle a, b, c\rangle = ae_1+be_2+ce_3$. As a programmer this is convenient because we can notate three real numbers in memory.
Recently, I've been looking at vectors and their kin from a different perspective. Like the classical way of viewing tensors, the vector is simply itself. There are operations I can do on it as a vector(like a dot product, or scalar product). These operations work perfectly without a basis. Its like everything I learned with a basis didn't need one at all!
What can I do with a vector if I have a basis that I can't do without one? The only operation I have found is "map a vector into 3 real numbers," which is useful to me as a programmer to encode a vector, but other than that, I'm having trouble finding what operations cannot be done without a basis.