📘 Chapter 3: Rank–Nullity Theorem

📖 Theorem Statement

Let \( T: V \rightarrow W \) be a linear transformation between finite-dimensional vector spaces. Then:

This is known as the Rank–Nullity Theorem, where:

• \( \dim(\ker(T)) \) = Nullity of \( T \)
• \( \dim(\text{Im}(T)) \) = Rank of \( T \)

🧠 Sketch of the Proof

The theorem relies on choosing a basis for the kernel and extending it to a basis for the domain space. Applying \( T \) to the extended basis yields a basis for the image.

🚀 Applications

• Determining if a matrix has a full-rank transformation
• Analyzing solutions to systems of equations
• Understanding dimensions of solution spaces

📊 Example

Let \( A = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \end{bmatrix} \) - Domain dimension = 3 (number of columns) - Row reduce to find Rank = 2 - So Nullity = \( 3 - 2 = 1 \)

📝 Practice

Q: A \( 4 \times 5 \) matrix has rank 3. What is its nullity?

Answer: Nullity = \( 5 - 3 = 2 \)