Rank and Nullity

Chapter 5 • Section 5-4

"The fundamental balance of linear algebra. ⚖️"

🏆 The Rank Theorem

For any $m \times n$ matrix $A$, the dimension of the row space equals the dimension of the column space.

$$ \dim(\text{row } A) = \dim(\text{col } A) = \rank(A) $$

This number is called the rank of the matrix.

🔍 Finding Bases

Given a matrix $A$, how do we find bases for its fundamental subspaces?

Basis for Row Space

1. Row reduce $A$ to REF.

2. The non-zero rows of the REF form a basis for $\text{row}(A)$.

Note: Use the rows of the REF, not the original matrix.

Basis for Column Space

1. Row reduce $A$ to REF.

2. Identify the pivot columns.

3. The corresponding columns in the original matrix $A$ form a basis for $\text{col}(A)$.

The Rank-Nullity Theorem

For any $m \times n$ matrix $A$ (which maps $\mathbb{R}^n \to \mathbb{R}^m$):

$$ \rank(A) + \nullity(A) = n $$

(Dimension of Image) + (Dimension of Kernel) = (Dimension of Domain)

💎 Full Rank Matrices

Full Column Rank ($\rank A = n$)

  • Columns are linearly independent.
  • $\nullity(A) = 0$ (Kernel is only $\{\vec{0}\}$).
  • $A\vec{x}=\vec{b}$ has at most one solution.

Full Row Rank ($\rank A = m$)

  • Rows are linearly independent.
  • $\text{col}(A) = \mathbb{R}^m$ (Columns span the whole codomain).
  • $A\vec{x}=\vec{b}$ has at least one solution for every $\vec{b}$.

Rank-Nullity Visualizer

Adjust the slider to change the Rank of a $3 \times 3$ matrix ($n=3$). See how Nullity changes to satisfy the theorem.

Rank 0 Rank 1 Rank 2 Rank 3
2
Rank
+
1
Nullity
=
3
n (Cols)

🧠 Knowledge Check Win $15

If a $5 \times 7$ matrix has rank 3, what is its nullity?