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Determine if the following is a linear transformation. If so, find the standard matrix of the transformation.

$T:P_1\to P_2$ such that $$T(p(t))= \int p(t) dt$$

I'd appreciate any help, an explanation of how to approach the problem would be extremely helpful. Thank you!

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There are no specifications for P1 and P2. What is stated in the problem is all the information give for those two values. –  Sharon Oct 18 '12 at 0:02
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Perhaps they are the spaces spanned by $\{1,x\}$ and $\{1,x,x^2\}$, respectively? Some books use that notation. –  sourisse Oct 18 '12 at 0:06
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@Sharon The use of $p$ strongly suggests polynomials, as sourisse is noting. Moreover, if you're being asked for the standard matrix, you need finitely generated vector spaces. –  Pedro Tamaroff Oct 18 '12 at 0:08
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I suspect it should be definite integral there. –  nikita2 Oct 18 '12 at 0:12
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Sharon, you find a basis $B$ for $P_1$, you find a basis $C$ for $P_2$, you apply $T$ to each element in $B$, you express each result in terms of $C$, you extract the coefficients of those expressions you have found, voila! there's your matrix. –  Gerry Myerson Oct 18 '12 at 1:35

2 Answers 2

Let's us use the standard basis for $P_1$, the vector space of polynomials of degree at most $1$. We will define the mapping $$T(p)=\int_0^x p(t)\ dt$$ notice that we need to take a definite integral to avoid ambiguity in the constant of integration.

Let us first prove that this mapping is linear. If we let $p$ and $q$ be polynomials in $P_1$ with scalar $c$, then we have $$T(cp + q) = \int_0^x cp(t) + q(t)\ dt = c\int_0^x p(t)\ dt + \int_0^x q(t)\ dt = cT(p) + T(q)$$ so the mapping is indeed linear.

If we feed the standard basis vectors into the mapping, we end up with $$T(1) = \int_0^x 1\ dt = x$$ $$T(t) = \int_0^x t\ dt = \frac{x^2}{2}$$ We can write the matrix of the mapping $T$ with respect to the standard basis vectors of $P_1$ and $P_2$ as $$[T] = \begin{pmatrix}0 & 0 \\ 1 & 0 \\ 0 & \frac{1}{2}\end{pmatrix}$$

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If your definition of a linear transformation matches Wikipedia, you need to prove that $T(kx)=kT(x)$ and $T(x+y)=T(x)+T(y)$. Both of these are usually proved properties of integrals, so go back to the proofs of these.

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