We are tasked with proving that for a linear map $T: V \to V$ on a finite-dimensional vector space $V$ over a field $F$, the condition $T^2 = 0$ (i.e., $T(T(v)) = 0$ for all $v \in V$) is equivalent to the condition $\text{range}(T) \subseteq \text{null}(T)$.

Forward Direction: $T^2 = 0 \implies \text{range}(T) \subseteq \text{null}(T)$

Proof:

Assume that $T^2 = 0$, meaning that for every $v \in V$, we have: $T(T(v)) = 0.$ This implies that the image of $v$ under $T$, denoted by $T(v)$, is mapped to 0 by $T$. In other words, for every $v \in V$, $T(v) \in \text{null}(T),$ where $\text{null}(T) = \{ v \in V \mid T(v) = 0 \}$.

Thus, $T(v) \in \text{null}(T)$ for every $v \in V$, meaning that all elements in the range of $T$ (i.e., all $T(v)$ for $v \in V$) are contained in $\text{null}(T)$. Therefore, we have: $\text{range}(T) \subseteq \text{null}(T).$

Reverse Direction: $\text{range}(T) \subseteq \text{null}(T) \implies T^2 = 0$

Proof:

Now, assume that $\text{range}(T) \subseteq \text{null}(T)$. This means that for every $v \in V$, $T(v) \in \text{null}(T),$ which implies that for every $v \in V$, $T(T(v)) = 0$.

Therefore, for all $v \in V$, $T^2(v) = T(T(v)) = 0,$ which is precisely the condition that $T^2 = 0$.

Conclusion:

We have shown both directions:

• If $T^2 = 0$, then $\text{range}(T) \subseteq \text{null}(T)$,
• If $\text{range}(T) \subseteq \text{null}(T)$, then $T^2 = 0$.

Thus, we conclude that $T^2 = 0$ if and only if $\text{range}(T) \subseteq \text{null}(T)$.

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Would you like further details or have any questions on this proof?

Here are 5 related questions to explore further:

1. What are the implications of $T^2 = 0$ on the eigenvalues of $T$?
2. How does the structure of $V$ change if $T^k = 0$ for some $k > 2$?
3. How can we generalize this result to infinite-dimensional vector spaces?
4. What does the Jordan canonical form of $T$ look like when $T^2 = 0$?
5. What other conditions can imply $T^2 = 0$ for a linear map $T$?

Tip: For any linear map $T$, understanding the interplay between the null space and the range can often provide key insights into the structure of the transformation.