Question

Let \( T : \mathbb{R}^7 \to \mathbb{R}^7 \) be a linear transformation with \( \text{Nullity}(T) = 2. \) Then, the minimum possible value for \( \text{Rank}(T^2) \) is ............

Hint:

For any linear map \( T \), null space enlarges under powers: \( N(T) \subseteq N(T^2) \subseteq N(T^3) \).

Answer 1

Correct Answer: 3

Step 1: Use the rank–nullity theorem. 
\[ \text{Rank}(T) + \text{Nullity}(T) = 7. \] \[ \Rightarrow \text{Rank}(T) = 7 - 2 = 5. \]

Step 2: Relationship between \( \text{Nullity}(T^2) \) and \( \text{Nullity}(T) \). 
\[ \text{Null}(T) \subseteq \text{Null}(T^2), \] so \( \text{Nullity}(T^2) \geq 2. \)

Step 3: For minimum possible \( \text{Rank}(T^2) \), take maximum nullity. 
Maximum \( \text{Nullity}(T^2) = 4 \) (since rank cannot increase). \[ \Rightarrow \text{Rank}(T^2) = 7 - 4 = 3. \]

Final Answer: \[ \boxed{3} \]