Are the following proofs correct (1)?

Question

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.

1. Show that a linear operator maps zero to zero.

My attempt:

Let $T$ be a linear operator such that $T:X\to Y$ and $0_x \in X$ and $0_y \in Y$ be the zero element of each space

Show $T(0_x) = y$, $y = 0_y$ and $y \in Y$

$$T(0_x)= T(0_x + 0_x) = T(0_x) + T(0_x) = y + y = y.$$

Thus $y = 0_y,$ as $y = y + y$ is only true for the zero element

2.Let $A: X \to Y$ be a linear operator from a linear space $X$ to a linear space $Y$. Show that the kernel of a linear operator $A: X \to Y$ is a linear subspace of $X$.

My attempt:

1. Showing non empty, $D : C ^∞ → C^∞$ , let D be the differential operator, then $\ker(D) = c$, $c \in F.$

2. Showing additive, where $x_1, x_2 \in \ker(X):$

$Ax_1 = 0$ and $Ax_2 = 0$

$A(x_1 + x_2) = 0$

3. Showing scalar multiplication for $a \in F:$

$a(Ax_1)=a0$
$a0 = a0$

Answer 1

1. Looks good to me. I'd maybe say one more sentence about how you know no nonzero vector satisfies $y+y=y$.

2.1. Why do you go to the differential operator here? You need to prove the kernel is always a linear subspace, not only in the case of the differential operator. In fact, why do you need to show at all that the kernel is nonempty?

2.2. Looks good. Perhaps note that what you did shows $x_1+x_2\in\ker A$.

2.3. I couldn't follow what you meant.

Answer 2

1: Rather than the airy "as $y+y=y$ is only true for the zero element". Just subtract $y$ from both sides to explicitly find $y$.

2.1: Why is there a differential operator here? You have already shown in problem 1 that the kernel is nonempty.