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The title might be misleading but I have forgotten how these functions are called. I am referring to cases such as:

$$7sin(y) + 0.9cos(y/4) = x_1sin(y) + x_2cos(y/4)$$

where we know that $x_1 = 7$ and $x_2 = 0.9$.

Another example would be polynomials:

$$5y^2 + 4y -12 = x_1y^2 + x_2y -x_3$$

where again the $x_i$ can be trivially found by equating the same-power terms.


Q: Generally speaking what conditions should, say $f(y)$ & $g(y)$ satisfy so that $x_1f(y) + x_2g(y) = 0$ requires $x_1 = 0$ and $x_2 = 0$?

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  • $\begingroup$ You need to make the question more precise if you want to get any help. What are some characteristics of these functions? Where did you hear about them?What makes the examples you gave orthogonal? $\endgroup$ – YiFan Sep 19 '18 at 12:41
  • $\begingroup$ @user496634 I hope it is clearer now. $\endgroup$ – Ev. Kounis Sep 19 '18 at 13:03
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Recall that two functions are defined orthogonal on $[a,b]$ if

$$\langle f,g\rangle=\int_a^b f(x)g(x)dx=0$$

but you example are referring to linear independence.

To show linear independence it suffices to show that

$$x_1f(y) + x_2g(y) = 0 \quad \forall y \implies x_1=x_2=0$$

that is, as an example, for the trigonometric functions

  • $y=0 \implies x_1\cdot 0 + x_2\cdot 1=0$
  • $y=\pi/2 \implies x_1 + x_2/\sqrt 2=0$

and from the two equations we obtain $x_1=x_2=0$.

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  • $\begingroup$ thanks for the answer. How is linear independence shown? How can one prove that two functions are linearly independent? $\endgroup$ – Ev. Kounis Sep 19 '18 at 13:00
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    $\begingroup$ @Ev.Kounis For the trigonometric functions note that at $y=0 \implies x_1\cdot 0 + x_2\cdot 1=0$ and $y=\pi/2 \implies x_1 + x_2/\sqrt 2=0$ and from the two equations we obtain $x_1=x_2=0$ $\endgroup$ – gimusi Sep 19 '18 at 13:12
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$\{ \sin(y), \cos(y/4)\}$ is linearly independent.

Since they are linearly independent

$$(7-x_1)\sin(y) + (0.9-x_2) \cos(y/4)=0$$ implies that $7-x_1=0$ and $0.9-x_2=0$.

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  • $\begingroup$ thanks for the answer. How is linear independence shown? How can one prove that two functions are linearly independent? $\endgroup$ – Ev. Kounis Sep 19 '18 at 13:00
  • $\begingroup$ it depends on case by case of the functions that you deal with. For example, to prove that $y^2$ and $y$ are independent, $ay^2+by=0, \forall y$, one possible reasoning is that a non-zero polynomial of degree $n$ has at most $n$ roots, but if every real number is a root, then it must be the zero polynomial. $\endgroup$ – Siong Thye Goh Sep 19 '18 at 13:05
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    $\begingroup$ A method to show linear independence of functions may be found in differential equations courses: The Wronskian determinant. en.wikipedia.org/wiki/Wronskian $\endgroup$ – GEdgar Sep 19 '18 at 13:06
  • $\begingroup$ Good point! I totally forget about Wronskian! $\endgroup$ – Siong Thye Goh Sep 19 '18 at 13:10
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$$x_1\,f(y)+x_2\,g(y)=0\implies x_1=x_2=0$$

can be proven by finding values of $y$, let $y_a,y_b$ such that

$$\begin{cases}x_1\,f(y_a)+x_2\,g(y_a)=0,\\x_1\,f(y_b)+x_2\,g(y_b)=0,\end{cases}$$ is a nonsingular linear system.

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