Question

Let $T : U(F) \to V(F)$ be a linear transformation, then which of the following holds:
A. $\text{Rank}(T) + \text{dim}(V) = \text{Nullity}(T)$ B. $\text{Rank}(T) + \text{Nullity}(T) = \text{dim}(U)$ C. $\text{Nullity}(T) = \text{dim}(V)$ D. $\text{Nullity}(T) = \text{dim}(\text{Ker}(T))$ E. $\text{Nullity}(T) = \text{dim}(U)$ Choose the correct answer from the options given below:

• A. B, C, E Only
• B. B, D Only
• C. A, D Only
• D. A, C Only

Hint:

Always remember: $\text{Rank}(T) + \text{Nullity}(T) = \text{dim}(\text{Domain})$. The codomain dimension $\text{dim}(V)$ does not appear in the theorem!

Answer 1

The Correct Option is B

Step 1: Concept:
This question tests basic definitions and the fundamental Rank-Nullity Theorem for linear transformations between vector spaces.

Step 2: Key Formula or Approach:
For a linear transformation $T : U \to V$ where $U$ is finite-dimensional:
1. $\text{Kernel of } T$: $\text{Ker}(T) = \{\mathbf{u} \in U \mid T(\mathbf{u}) = \mathbf{0}\}$.
2. $\text{Nullity of } T$: $\text{Nullity}(T) = \text{dim}(\text{Ker}(T))$.
3. $\text{Rank of } T$: $\text{Rank}(T) = \text{dim}(\text{Im}(T))$.
4. $\text{Rank-Nullity Theorem}$: \[ \text{Rank}(T) + \text{Nullity}(T) = \text{dim}(U) \]

Step 3: Step-by-step Explanation:

• Statement A: $\text{Rank}(T) + \text{dim}(V) = \text{Nullity}(T)$ is incorrect as per Rank-Nullity Theorem.

• Statement B: $\text{Rank}(T) + \text{Nullity}(T) = \text{dim}(U)$ is the exact statement of the Rank-Nullity Theorem. Hence, Statement B is correct.

• Statement C: $\text{Nullity}(T) = \text{dim}(V)$ is generally false; Nullity relates to the domain $U$, not the codomain $V$.

• Statement D: By definition, the dimension of the kernel of $T$ is called the nullity of $T$. Hence, Statement D is correct.

• Statement E: $\text{Nullity}(T) = \text{dim}(U)$ holds only if $T$ is the zero transformation. Hence, it is not true in general.

Step 4: Final Answer:
Only statements B and D are correct. Therefore, option (B) is the correct answer.