Let V be a finite-dimensional vector space over a field F and T : V → V
a linear map. Prove that T
2 = 0 (where 0 denotes the zero map V → V ) if and only if
range(T) ⊂ null(T).

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 \( \text{range}(T) \subseteq \text{null}(T) \).

---

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.

Let V be a finite-dimensional vector space over a field F and T : V → V a linear map. Prove that T^2 = 0 (where 0 denotes the zero map V → V) if and only if range(T) ⊂ null(T).

Explore the proof that for a linear map T on a finite-dimensional vector space V, T^2 = 0 is equivalent to the condition that the range of T is a subset of its null space. This problem involves key concepts in linear algebra, such as vector spaces, linear transformations, and the relationship between range and null space.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation