# Linear transformations of the real polynomial space

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.

• "Matricidal" sounds like someone who murdered a matrix... Jan 6 '14 at 5:29
• Matrix representation sorry. Jan 6 '14 at 5:30
• 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 $X$. Jan 6 '14 at 5:40

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.