Can $u_n$ be a linear combination of the vectors in infinite set $S$?

Question

Suppose $u_n = a_1u_1 + a_2u_2$ for vectors $u$ in infinite set $S$. Even though it would seem to be, can $u_n$ be said to be a linear combination of the vectors in $S$? Because presumably that would mean to say $$u_n = a_1u_1 + a_2u_2 + 0u_3 + ...$$ But my textbook defines a vector $u_n$ to be a linear combination of the vectors in $S$ if there exists a finite number of vectors and scalars such that you have equality. But here, technically, you do not have a finite number of vectors and scalars.

Answer 1

Yes, you can find many examples of such linearly independent vectors.

For example, $$1,x,x^2, x^3,....$$

$$ \sin x, \sin 2x, \sin 3x, ......$$

Orthogonal polynomials are also good examples of linearly independent sets.