Differential Equation Linearly İndependent

Question

İf u₁(x) and u₂(x) linearly independent, you prove it linearly independent A₁u₁(x)+A₂u₂(x) and B₁u₁(x)+B₂u₂(x)

(A₁B₂-A₂B₁ not equal 0)

Answer 1

Start with the definition of linear independence:

Because u1 and u2 are l.i., it follows that the only solution to c1*u1+c2*u2 = 0 is (c1,c2) = (0,0).

Now, set up the linear combination

d1*(A1u1+A2u2)+d2*(B1u1+B2u2) = 0 and show that (d1,d2) must be (0,0).