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2h2o
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Homework Statement
Show directly that the given functions are linearly dependent on the real line==>find a non-trivial linear combination of the given functions that vanishes identically.
Homework Equations
[tex]f(x) = 2x, g(x) = 3x^{2}, h(x) = 5x-8x^{2}[/tex]
The Attempt at a Solution
I know that the structure of the solution will be [tex]C_{1}y_{1}+C_{2}y_{2}+C_{3}y_{3}=0[/tex], so I find three equations for the three unknown constants.
[tex]C_{1}2x + C_{2}3x^{2} + C_{3}(5x-8x^{2}) = 0 [/tex]...[1]
first derivative
[tex]2C_{1} + 6C_{2}x + C_{3}(5-16x) = 0 [/tex]...[[2]
second derivative
[tex]0 + 6C_{2} -16C_{3} = 0 [/tex]...[3]
Upon which I begin finding the constants:
Equation [3] immediately shows that [tex]6C_{2}=16C_{3}[/tex]
Then [2] becomes
[tex]2C_{1}+5C_{3}=0[/tex]
hence [tex]2C_{1} = -5C_{3}[/tex]
Now [tex]C_{1}=\frac{-5C_{3}}{2}[/tex]
So [tex]C_{1}=\frac{-5}{2}*\frac{3C_{2}}{8} = \frac{-15}{16}C_{2}[/tex]
Now everything I try gives me the wrong solutions..matrix, substitution (infinite substitutions since every constant is related to another constant.)
So how do I prove that these functions are L.D.?
Magically (or so it seems) the solution is supposed to be [tex]15(2x) - 16(3x^{2}) - 6(5x-8x^{2})=0[/tex]