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Let n a natural fixed number and X the space of all real polynomials of degree at most n. I need to give a basis for X and say what of these following transformations are linear in X in X, this is an exercise from linear algebra but i can't solve it.

$p(x) \rightarrow{} \frac{dp(x)}{dx} + x$ , $p(x) \rightarrow{} \int_0^x p(y) dy$.

First i try put the matrix representation but i can't , please can you help me.Thxs.And nice year for all.

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  • $\begingroup$ "Matricidal" sounds like someone who murdered a matrix... $\endgroup$
    – DonAntonio
    Jan 6 '14 at 5:29
  • $\begingroup$ Matrix representation sorry. $\endgroup$
    – Rachel
    Jan 6 '14 at 5:30
  • $\begingroup$ The first transformation is not linear. For example, the transformation takes $1$ to $x$, and also $2$ to $x$. The second is linear, but not from $X$ to $X$. For let $n=3$ say. Then $4x^3$ is taken to $x^4$, which is not in $X$. $\endgroup$ Jan 6 '14 at 5:40
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I will show for $n=2$, you can work the general case.

As basis pick $b_1=1$, $b_2=x$, $b_3=x^2$. As you can see that any quadratic polynomial is a linear combination of these.

The derivative operator gives $$ D(b_1)=0 \\D(b_2)=b_2\\ D(b_3) = 2 b_2$$ $$ So the derivative is linear.

The operator that gives $+x$ is not linear (it is not even a function of its input). So $p$ is not linear.

Integration takes you out of the space. If you permit that then you need one more basis for the range. Let it be $b_4 = x^3$. Then the integrator, $I$, is: $$ I(b_1) = b_2\\I(b_2) = b_3/2\\I(b_3) = b_4/3$$ and is clearly linear.

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  • $\begingroup$ Thanxs, iw ill try for n=3 and then in general . $\endgroup$
    – Rachel
    Jan 6 '14 at 6:23

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