Determine whether the given set of functions is linearly independent on the interval (-∞, ∞). f1(x)=0, f2(x)=x, f3(x)=e^x | Numerade
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Let ϕ1, ϕ2 be two differentiate functions on an interval I, which are not necessarily solutions of an equation L(y)=0. Prove the following: (a) If ϕ1, ϕ2 are linearly dependent on I, then W(ϕ1, ϕ2)(x)=0 for all x in I. (b) If W(ϕ1, ϕ2)(x0) ≠0 for some x0 in I; then ϕ1, ϕ2 are linearly independent on I (c) W(ϕ1, ϕ2)(x)=0 for all x in I does not imply that ϕ1, ϕ2 are linearly dependent on I. (Hint: Ex. 3.) (d) W(ϕ1, ϕ2)(x)=0 for all x in I, and ϕ2(x) ≠0 on I, imply that ϕ1, ϕ2 are linearly depende
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