Question

If \( T: V \to W \) is a linear transformation, what is the relationship between rank(\(T\)), nullity(\(T\)), and dim(\(V\))?

• A. rank(\(T\)) + nullity(\(T\)) = dim(\(W\))
• B. rank(\(T\)) $\times$ nullity(\(T\)) = dim(\(V\))
• C. rank(\(T\)) + nullity(\(T\)) = dim(\(V\))
• D. rank(\(T\)) = nullity(\(T\))

Hint:

Always remember: \[ \text{Rank} + \text{Nullity} = \text{Dimension of domain} \]

Answer 1

The Correct Option is C

Concept: Rank–Nullity Theorem
This is a fundamental theorem in linear algebra which states: \[ \text{rank}(T) + \text{nullity}(T) = \dim(V) \]
Step 1: Understand terms

• Rank: Dimension of the image (range) of \(T\)
• Nullity: Dimension of the kernel (null space) of \(T\)
• dim(\(V\)): Dimension of domain space

Step 2: Interpretation
The theorem splits the domain space into:

• Part mapped to zero (kernel)
• Part mapped to image

Step 3: Conclusion
\[ \text{rank}(T) + \text{nullity}(T) = \dim(V) \]