Hello, there is this question in the book:(adsbygoogle = window.adsbygoogle || []).push({});

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Consider the vector space of functions defined for t>0. Show that the following pairs of functions are linearly independent.

(a) t, 1/t

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So if they are linearly independent then there are a,b inR, such that

at + b/t = 0

So if we differentiate both sides with respect to t we get [tex]a + \frac{-b}{t^2} = 0[/tex]

which implies [tex]a = \frac{b}{t^2}[/tex]

If we plug this into the first equation, we get [tex] \frac{b}{t^2}t + \frac{b}{t} = 0[/tex]

so, [tex]\frac{2b}{t} = 0 => b = 0[/tex]

If we plug this back to the first equation it follows that a = 0. So a = 0 and b = 0, and therefore the two functions, t and 1/t, are linearly independent.

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Is that sufficient?

Also, what if I just said something like let t = 1, you then get the equation

a + b = 0

then let t = 2, and you get the equation

2a + b/2 = 0

And if you solve these equations simultaneously, it follows that a = 0 and b = 0 are the only solutions. Would that be sufficient, given that we know a = 0 and b = 0 are always solutions?

Thanks!

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# Linear Independence of two Functions

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