- #1
Chris1557
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Homework Statement
From Calculus we know that, for any polynomial function f : R -> R of degree <= n, the function
I(f) : R -> R, s -> ∫0s f(u) du, is a polynomial function of degree <= n + 1.
Show that the map
I : Pn -> Pn+1; f -> I(f),
is an injective linear transformation, determine a basis of the image of I and fi nd the matrix
M in M(n+2)x(n+1)(R) that represents I with respect to the basis 1,t,...,tn of Pn and the basis 1,t,...,tn+1 of Pn+1.
Homework Equations
The Attempt at a Solution
I found a topic involving setting f(x) = anxn + an-1xn-1 + an-2xn-2 + ... + a1x + a0, and g(x) = bnxn + bn-1xn-1 + bn-2xn-2 + ... + b1x + b0.
Then show that L(f + g) = L(f) + L(g) and that aL(f) = L(ax).
That seems ok, but I have no idea how to determine a basis of the image and I am confused on what the final part in notating and what to do.
Thanks.