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.

  • $\begingroup$ "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

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.

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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.