This question asks for the statement of one of the most fundamental theorems in linear algebra, which connects the dimensions of the key vector spaces associated with a linear transformation.
Step 1: Understanding the Question:
We need to find the correct formula that relates the dimensions of the domain, the image (range), and the kernel (null space) of a linear transformation \(T\) from a vector space \(V\) to a vector space \(W\).
Step 2: Key Formula or Approach:
This relationship is given by the Rank-Nullity Theorem. We need to recall its precise statement.
Step 3: Detailed Explanation:
Let's define the terms for a linear transformation \(T: V \to W\):
dim(V): The dimension of the domain space (the input space).
rank(T): The dimension of the image of \(T\), which is the set of all possible outputs in \(W\). It is a subspace of \(W\). rank(T) = dim(Im(T)).
nullity(T): The dimension of the kernel (or null space) of \(T\), which is the set of all vectors in \(V\) that are mapped to the zero vector in \(W\). It is a subspace of \(V\). nullity(T) = dim(Ker(T)).
The Rank-Nullity Theorem states that the dimension of the domain is equal to the sum of the dimension of the image and the dimension of the kernel.
\[ \text{rank}(T) + \text{nullity}(T) = \dim(V) \]
The theorem essentially says that the dimension of the input space is split between the dimensions of the part that gets "squashed" to zero (the kernel) and the part that effectively forms the output (the image).
Step 4: Final Answer:
The correct relationship is rank(\(T\)) + nullity(\(T\)) = dim(\(V\)).