Question

Are Sin(x) and e^x linearly independent? Justify.

Answer 1

yes they are linearly independent

Explanation:

By definition of linear dependence we have if f(x) and g(x) be 2 functions

if they are linearly dependent then we can write

f(x) = αg(x)...........(i) where α is an arbitrary constant

in our case we can see that the range of sin(x) is only from [-1,+1] while as
`e^(x)` has range from [0,∞]

thus we cannot find any value of α for which (i) is valid