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.
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.
