I'm hoping I can get some help with the following question:(adsbygoogle = window.adsbygoogle || []).push({});

Does definite integration (from x = 0 to x = 1) of functions in P_{n}correspond to some linear transformation from R^{n+1}to R?

OK, well my original answer was yes, but the textbook says "no, except for P_{0}" which I do not understand.

So I have p(x) = a_{n}x^{n}+ a_{n-1}x^{n-1}+ ... + a_{1}x + a_{0}

If I integrate from 0 to 1, I get: P(1) = a_{n}/(n+1) + a_{n-1}/n + ... + a_{1}/2 + a_{0}

Right?

So I have T(a_{n}, a_{n-1}, ..., a_{1}, a_{0}) = (a_{n}/(n+1) + a_{n-1}/n + ... + a_{1}/2 + a_{0})

and T: R^{n+1}-> R

And if I want to show that it is linear, then I show that the transformation has these properties:

T(kx) = kT(x) and

T(x+y) = T(x) + T(y)

and I think both cases are obvious.

And T is multiplication by [1/(n+1) | 1/n . . . 1/2 | 1]

What am I missing?

Thanks in advance for any help.

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# Homework Help: Linear transformation

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