The Row Space & The Left Null Space

Completing the Picture of the Four Fundamental Subspaces

We have spent a great deal of time focusing on the **columns** of our matrix `A`, which gave us the **Column Space** (`C(A)`) and the **Null Space** (`N(A)`).

But what about the **rows**? The rows of a matrix are vectors too. By exploring the world of the rows, we will uncover the final two pieces of our puzzle: the **Row Space** and the **Left Null Space**.

The Row Space: The World of 'Effective' Inputs

The **Row Space** of a matrix `A`, written as `C(Aᵀ)`, is the **span of the row vectors of `A`**.

It represents the space of **"effective" inputs**—the part of the input space that the matrix `A` actually "pays attention to." Any part of an input vector that is orthogonal to the Row Space will be "ignored" by the transformation and sent to zero. In fact, the Row Space is the **orthogonal complement** of the Null Space.

Key Idea

The matrix `A` acts as a perfect, one-to-one mapping from its Row Space to its Column Space.

Finding a Basis for the Row Space

The **non-zero rows of the Row Echelon Form (REF)** of `A` form a basis for the Row Space of `A`.

For our familiar matrix `A` and its REF, `U`:

A = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 2 & 4 & 7 & 9 \\ 3 & 6 & 8 & 11 \end{bmatrix} \quad \xrightarrow{} \quad U = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \end{bmatrix}

A basis for the Row Space of `A` is:

\{ [1, 2, 3, 4], [0, 0, 1, 1] \}

Profound Result

The dimension of the Row Space is always equal to the dimension of the Column Space.
dim(Row Space) = dim(Column Space) = rank(A)

The Left Null Space: The Final Piece

The **Left Null Space** of `A` is the Null Space of `Aᵀ`. It is the set of all vectors `y` such that `Aᵀy = 0`.

It's called the "Left" Null Space because if we transpose the equation `Aᵀy = 0`, we get `yᵀA = 0`, so the vector `y` is on the *left* of `A`.

The Left Null Space is the orthogonal complement to the Column Space.

The Grand Unification: The Fundamental Theorem of Linear Algebra

For any `m x n` matrix `A` with rank `r`, the four fundamental subspaces have the following properties and relationships.

Subspace	Lives In	Dimension
Column Space $C(A)$	Output Space ( $\mathbb{R}^m$ )	`r`
Null Space $N(A)$	Input Space ( $\mathbb{R}^n$ )	`n - r`
Row Space $C(A^T)$	Input Space ( $\mathbb{R}^n$ )	`r`
Left Null Space $N(A^T)$	Output Space ( $\mathbb{R}^m$ )	`m - r`

This is the big picture. This is the complete, elegant, geometric theory of what any matrix `A` *is* and *does*. It takes the entire input space, splits it into two perpendicular subspaces (Row Space and Null Space), and transforms one into the Column Space while annihilating the other.

Lesson 2.7: The Null Space

Lesson 2.9: The Fundamental Theorem of Linear Algebra