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Row Rank

We already know that the rank of a matrix $A$ is the dimension of its column space. Now we define the row rank as the rank of $A^t$ , which equals the dimension of the span of the row vectors of $A$ .

Think of each row of $A$ as a $1 \times n$ row vector. These row vectors span a subspace of $\mathbb{R}^n$ — the row space of $A$ . The row rank is the dimension of this row space, i.e., the number of linearly independent rows.

At first glance, there is no reason to expect row rank and column rank to be equal. Rows and columns have completely different dimensions. But the Row Rank Theorem tells us they are always equal — a deep and non-obvious symmetry of every matrix.

Formal View

Definition 5.3 — Row Rank

Let

A

be an

m \times n

matrix. Its row rank is defined as

\text{Rowrank}(A) := \text{Rank}(A^t)

, the dimension of the span of the row vectors of

A

Theorem 5.4 — Row Rank Theorem

For any matrix

C

\text{Rank}(C) = \text{Rowrank}(C)

The column rank and row rank of every matrix are equal.

The proof uses matrix factoring — we will build it up in the next section.

Interactive Visualization

Transpose Visualizer

Why This Matters

The equality of row rank and column rank reveals a fundamental symmetry hidden inside every matrix.

In data science, the rank of a data matrix counts the number of truly independent features regardless of whether we view data as rows or columns
The theorem guarantees that the number of independent equations in a linear system equals the number of independent constraints — rows and columns carry the same information about rank
It underpins the singular value decomposition: the non-zero singular values of $A$ and $A^t$ are identical

Learning Resources

Column space and row space

3Blue1Brown

Intuition for how column and row spaces are related.

10 min

Rank and the row reduced form

MIT OpenCourseWare

Gilbert Strang on rank, column space, and row space.

45 min

Quiz

Question 1

A $4 \times 6$ matrix $A$ has $\text{Rank}(A) = 3$ . What is $\text{Rank}(A^t)$ ?

Question 2

A matrix can have more linearly independent rows than linearly independent columns.

Common Mistakes

Thinking row rank and column rank could differ — they are always equal, which is the content of the Row Rank Theorem.
Confusing the row space (a subspace of $\mathbb{R}^n$ ) with the column space (a subspace of $\mathbb{R}^m$ ) — they live in completely different spaces.
Assuming the row rank of a tall matrix ( $m > n$ ) must be larger because there are more rows — rank is bounded by $\min(m, n)$ regardless.