A list of some questions from class and my attempt at them are given below. Would appreciate any advice on what I did wrong or on how to attempt some of the questions.
- Let A: X → Y be a linear operator from a linear space X to a linear space Y. Show that the image of a linear operator A: X → Y is a linear subspace of Y.
My attempt:
1.Showing non empty, I don't know any examples..
2.Showing additive, where $y_1, y_2 \in \ker(A):$
$Ax_1 = y_1$ and $Ax_2 = y_2$
$A(x_1 + x_2) = y_1 + y_2$
3.Showing scalar multiplication for $a \in F$
$a(Ax_1)=ay_1$
$ay_1 = ay_1$
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4.(a) The differentiation operator $D : C^∞ → C^∞$ is a linear operator. Find the kernel and image of D. (b) The operator $D^n : C^∞ → C^∞$ is a linear operator (differentiation n times). Find the kernel and image of $D^n$ .
My attempt:
(a) ker(D) = c, c ∈ F and im(D) = $C^∞$ , I don't know how to prove that this is the case it just seems like it is such.
(b) ker($D^n$) = $x^n$, x ∈ $C^∞$ and im($D^n$) = $C^∞$ , I don't know how to prove that this is the case it just seems like it is such.
3.1: See problem 1.
4(a): It is a standard application of the mean value theorem that $Df = 0$ implies $f$ is constant.
4(b): Your kernel is incorrect. What can you say about all polynomials of degree $<n$? For the image, what is the definition of $C^\infty$?