Question

what is the difference between linearly independence and linearly dependence and how we tell the difference?

Answer 1

So that means when it's independent , there is no free variable??

Answer 2

Let f(x) and g(x) be functions. For f(x) and g(x) to be linearly independent we must have
a*f(x) + b*g(x) = 0 if and only if a=b=0.
In other words there is no c such that a or b is not zero but
a*f(c) + b*g(c) = 0
If there is such a c, then we say that f(x) and g(x) are linearly dependent.

e.g.
f(x) = sin(x) and g(x) = cos(x) are linearly independent
f(x) = sin(x) and g(x) = sin(2x) are not linearly dependent (Why?)

Answer 3

because sin(2x) is a multiable of sin(x)??