The Equivalence Condition When the Range of a Linear Transformation is Smaller than the Kernel
Theorem1
Let $V$ be a vector space, and $T : V \to V$ be a linear transformation. Then, the following holds:
$$ T^{2} = T_{0} \iff R(T) \subset N(T) $$
Here, $T_{0}$ is the zero transformation, and $R(T), N(T)$ are the respective range and null space of $T$.
Generalization
Let $U, V, W$ be a vector space, and $T_{1} : U \to V$, $T_{2} : V \to W$ be linear transformations. Then, the following holds:
$$ T_{2}T_{1} = T_{0} \iff R(T_{1}) \subset N(T_{2}) $$
Explanation
It’s actually an obvious matter if you think about it. The proof method for the generally written theorem is the same.
Meanwhile, a linear transformation that is $T^{2} = T_{0}$ is called Nilpotent.
Proof
$(\Longrightarrow)$
Let’s assume $T^{2} = T_{0}$. Let $T(x) \in R(T)$ $(x\in V)$. Then,
$$ T(T(x)) = T^{2}(x) = 0 $$
Therefore, by the definition of $N(T)$, $T(x) \in N(T)$ is true. Hence,
$$ R(T) \subset N(T) $$
$(\Longleftarrow)$
Let’s assume $R(T) \subset N(T)$. Then, for all $x \in V$, since $T(x) \in R(T) \subset N(T)$, by the definition of $N(T)$,
$$ R(T) \subset N(T) $$
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Stephen H. Friedberg, Linear Algebra (4th Edition, 2002), p97 ↩︎