Linear combination
Functions

Let K be the set C of all complex numbers, and let V be the set C (R) of all continuous functions from the real line R to the complex plane C.Consider the vectors (functions) f and g defined by f (t) := e and g (t) := e.(Here, e is the base of the natural logarithm, about 2.71828..., and i is the imaginary unit, a square root of −1.)Some linear combinations of f and g are:

On the other hand, the constant function 3 is not a linear combination of f and g. To see this, suppose that 3 could be written as a linear combination of e and e. This means that there would exist complex scalars a and b such that ae + be = 3 for all real numbers t. Setting t = 0 and t = π gives the equations a + b = 3 and a + b = −3, and clearly this cannot happen. See Euler's identity.
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