As evidenced by the confusion of at least one commenter, I do not think it is a good didactic way to introduce vectors by how they can be written in a particular basis.
It is just unhelpful in many ways. It fixates on one particular basis and it results in a vector space with few applications and it can not explain many of the most important function vector spaces, which are of course the L^p spaces.
In most function vector spaces you encounter in mathematics, you can not say what the value of a function at a point is. They are not defined that way.
The right didactic way, in my experience, is introducing vector spaces first. Vectors are elements of vector spaces, not because they can be written in any particular basis, but because they fulfill the formal definition. And because they fullfil the formal definition they can be written in a basis.
It is just unhelpful in many ways. It fixates on one particular basis and it results in a vector space with few applications and it can not explain many of the most important function vector spaces, which are of course the L^p spaces.
In most function vector spaces you encounter in mathematics, you can not say what the value of a function at a point is. They are not defined that way.
The right didactic way, in my experience, is introducing vector spaces first. Vectors are elements of vector spaces, not because they can be written in any particular basis, but because they fulfill the formal definition. And because they fullfil the formal definition they can be written in a basis.