My teacher has this exercise in his notes:
$$\text{Let }T\in\mathcal{L}(V).\text{ Prove that }T^2=0\Leftrightarrow\text{Im}(T)\subseteq\text{Ker}(T).\tag{*}\label{*}$$ where $\mathcal{L}(V)$ is the set of all linear maps $V\to V$, and $\text{Ker}(T)=${$x\in V\mid T(x)=0$}. My teacher did not specify what the $0$ in $\eqref{*}$ means. Can anyone help me understand what he might mean with this symbol?
On
In the context of linear maps, $0$ represents the zero vector, denoted as $(\mathbf{0})$, which is the additive identity element in the vector space $V$. It is the vector that satisfies the property $(\mathbf{v} + \mathbf{0} = \mathbf{v})$ for any vector $(\mathbf{v})$ in $V$.
In the given exercise, $T^2 = \mathbf{0}$ means that the composition of the linear map $T$ with itself results in the zero map, denoted as $(\mathbf{0})$, which maps every vector in $V$ to the zero vector.
$Im(T)$ represents the image of the linear map $T$, the subspace consisting of vectors that $T$ can map to. $Ker(T)$ represents the kernel of $T$, the subspace consisting of vectors that $T$ maps to the zero vector.
The statement $T^2 = \mathbf{0}$ implies $Im(T) \subseteq Ker(T)$ means that if $T$ composed with itself is the zero map, then every vector in the image of $T$ is also in the kernel of $T$. In other words, the image of $T$ is a subset of the kernel of $T$.
To prove this, assume $T^2 = \mathbf{0}$ and show that for any vector $x$ in $Im(T)$, $T(x) = \mathbf{0}$. This demonstrates that the image of $T$ is indeed contained within the kernel of $T$.
It's always recommended to clarify any symbol or notation ambiguity with your teacher for a precise understanding of their intended meaning.
HOPE THAT HELPS
In the statement “$T^2=0\iff\operatorname{Im}(T)\subseteq\operatorname{Ker}(T)$”, the symbol $0$ refers to the zero map $\phi:V\to V$ such that $\phi(v)=\vec 0$ for all $v\in V$, where $\vec 0$ is the additive identity element of the vector space $V$.
This notation might seem injudicious to you, but it is not without merit. In any vector space, it is common to denote its additive identity element by $0$, even when the elements of that vector space happen to be functions, sets, matrices, etc. rather than elements of (say) $\mathbb R^n$. Why is this relevant? If $V$ is a vector space over a field $K$, then $L(V)$ is itself a vector space over $K$, and the zero map is its additive identity. As a result, it is customary to denote this map by $0$. If there are multiple vector spaces under consideration, then it is usually clear from context which $0$ is meant. For instance, in your case, $T^2$ is the linear map obtained by composing $T$ with itself, and so it would be absurd for $0$ to denote the additive identity of $V$, rather than $L(V)$; in general, the members of a vector space $V$ are not linear maps.