MIT 18.06SC Lecture 10: Four Fundamental Subspaces

linear algebra

MIT 18.06

subspaces

rank

The four fundamental subspaces that completely characterize any matrix: column space, null space, row space, and left null space

Author

Chao Ma

Published

October 21, 2025

Overview

For any $m \times n$ matrix $A$, there are four fundamental subspaces that completely characterize its structure:

Subspace	Notation	Lives in	Dimension	Solves
Column Space	$C(A)$	$\mathbb{R}^m$	$r$	$Ax = b$
Null Space	$N(A)$	$\mathbb{R}^n$	$n - r$	$Ax = \mathbf{0}$
Row Space	$C(A^T)$	$\mathbb{R}^n$	$r$	$A^T y = c$
Left Null Space	$N(A^T)$	$\mathbb{R}^m$	$m - r$	$A^T y = \mathbf{0}$

where $r = \text{rank}(A)$.

Column Space: $C(A)$

Definition: The span of all columns of $A$.

Properties:

Lives in $\mathbb{R}^m$ (same dimension as the rows)
Dimension = $\text{rank}(A) = r$
Represents all possible outputs of $Ax$

Linear combination interpretation:

\[ x_1 A_{:,1} + x_2 A_{:,2} + \cdots + x_n A_{:,n} = b \]

Key question: For which $b$ does $Ax = b$ have a solution?

Answer: When $b \in C(A)$ (when $b$ is in the column space).

Null Space: $N(A)$

Definition: All vectors $x$ such that $Ax = \mathbf{0}$.

Properties:

Lives in $\mathbb{R}^n$ (same dimension as the columns)
Dimension = $n - r$ (number of free variables)
Contains all solutions to the homogeneous system $Ax = \mathbf{0}$

Interpretation: Vectors that get “killed” by the matrix $A$.

Row Space: $C(A^T)$

Definition: The span of all rows of $A$, equivalently the column space of $A^T$.

Properties:

Lives in $\mathbb{R}^n$ (same dimension as the columns)
Dimension = $\text{rank}(A) = r$ (same as column space)
Represents all possible outputs of $A^T y$

Key Insight

Row rank equals column rank (both equal $r$).

Left Null Space: $N(A^T)$

Definition: All vectors $y$ such that $A^T y = \mathbf{0}$.

Properties:

Lives in $\mathbb{R}^m$ (same dimension as the rows)
Dimension = $m - r$
Also called the left null space of $A$

Why “left null space”?

Starting from $A^T y = \mathbf{0}$:

\[ \begin{aligned} A^T y &= \mathbf{0} \\ (A^T y)^T &= \mathbf{0}^T = [\mathbf{0}] \\ y^T A &= [\mathbf{0}] \end{aligned} \]

Since $y^T$ is $1 \times m$ and appears on the left of $A$, this is called the left null space.

RREF and the Four Subspaces

Example Matrix

\[ A = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \\ 2 & 4 & 6 & 8 \end{bmatrix} \]

Row Echelon Form (REF)

\[ \text{REF} = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 0 & -4 & -8 & -12 \\ 0 & 0 & 0 & 0 \end{bmatrix} \]

Reduced Row Echelon Form (RREF)

Step 1: Normalize pivot rows

\[ \begin{bmatrix} 1 & 2 & 3 & 4 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 0 & 0 \end{bmatrix} \]

Step 2: Eliminate above pivots

\[ \text{RREF} = \begin{bmatrix} 1 & 0 & -1 & -2 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 0 & 0 \end{bmatrix} \]

Analysis

Rank: $r = 2$ (two pivot columns)
Free variables: $n - r = 4 - 2 = 2$ (columns 3 and 4)
Dimension of $N(A)$: $n - r = 2$
Dimension of $C(A)$: $r = 2$
Dimension of $C(A^T)$: $r = 2$
Dimension of $N(A^T)$: $m - r = 3 - 2 = 1$

Key Observation

RREF is the unique simplest form of a matrix obtained through row operations that preserve the solution space.

Visualization

Dimension Relationships

For an $m \times n$ matrix $A$ with rank $r$:

In $\mathbb{R}^n$:

$\dim(C(A^T)) + \dim(N(A)) = r + (n - r) = n$
Row space and null space partition $\mathbb{R}^n$

In $\mathbb{R}^m$:

$\dim(C(A)) + \dim(N(A^T)) = r + (m - r) = m$
Column space and left null space partition $\mathbb{R}^m$

Key Insight

The four subspaces come in complementary pairs that completely partition their ambient spaces.

Summary

The four fundamental subspaces:

Column space $C(A)$: where outputs $Ax$ live
Null space $N(A)$: inputs that map to zero
Row space $C(A^T)$: perpendicular complement to null space in $\mathbb{R}^n$
Left null space $N(A^T)$: perpendicular complement to column space in $\mathbb{R}^m$

Dimension formula:

$\dim(C(A)) = \dim(C(A^T)) = r$
$\dim(N(A)) = n - r$
$\dim(N(A^T)) = m - r$

Total check: $r + (n-r) = n$ and $r + (m-r) = m$

Source: MIT 18.06SC Linear Algebra, Lecture 10

--- title: "MIT 18.06SC Lecture 10: Four Fundamental Subspaces" author: "Chao Ma" date: "2025-10-21" categories: [linear algebra, MIT 18.06, subspaces, rank] description: "The four fundamental subspaces that completely characterize any matrix: column space, null space, row space, and left null space" --- {{< video https://www.youtube.com/watch?v=nHlE7EgJFds >}} ## Overview For any $m \times n$ matrix $A$, there are **four fundamental subspaces** that completely characterize its structure: | Subspace | Notation | Lives in | Dimension | Solves | |----------|----------|----------|-----------|--------| | **Column Space** | $C(A)$ | $\mathbb{R}^m$ | $r$ | $Ax = b$ | | **Null Space** | $N(A)$ | $\mathbb{R}^n$ | $n - r$ | $Ax = \mathbf{0}$ | | **Row Space** | $C(A^T)$ | $\mathbb{R}^n$ | $r$ | $A^T y = c$ | | **Left Null Space** | $N(A^T)$ | $\mathbb{R}^m$ | $m - r$ | $A^T y = \mathbf{0}$ | where $r = \text{rank}(A)$. ## Column Space: $C(A)$ **Definition**: The span of all columns of $A$. **Properties**: - Lives in $\mathbb{R}^m$ (same dimension as the rows) - Dimension = $\text{rank}(A) = r$ - Represents all possible outputs of $Ax$ **Linear combination interpretation**: $$ x_1 A_{:,1} + x_2 A_{:,2} + \cdots + x_n A_{:,n} = b $$ **Key question**: For which $b$ does $Ax = b$ have a solution? **Answer**: When $b \in C(A)$ (when $b$ is in the column space). ## Null Space: $N(A)$ **Definition**: All vectors $x$ such that $Ax = \mathbf{0}$. **Properties**: - Lives in $\mathbb{R}^n$ (same dimension as the columns) - Dimension = $n - r$ (number of free variables) - Contains all solutions to the homogeneous system $Ax = \mathbf{0}$ **Interpretation**: Vectors that get "killed" by the matrix $A$. ## Row Space: $C(A^T)$ **Definition**: The span of all rows of $A$, equivalently the column space of $A^T$. **Properties**: - Lives in $\mathbb{R}^n$ (same dimension as the columns) - Dimension = $\text{rank}(A) = r$ (same as column space) - Represents all possible outputs of $A^T y$ ::: {.callout-tip} ## Key Insight Row rank equals column rank (both equal $r$). ::: ## Left Null Space: $N(A^T)$ **Definition**: All vectors $y$ such that $A^T y = \mathbf{0}$. **Properties**: - Lives in $\mathbb{R}^m$ (same dimension as the rows) - Dimension = $m - r$ - Also called the **left null space** of $A$ **Why "left null space"?** Starting from $A^T y = \mathbf{0}$: $$ \begin{aligned} A^T y &= \mathbf{0} \\ (A^T y)^T &= \mathbf{0}^T = [\mathbf{0}] \\ y^T A &= [\mathbf{0}] \end{aligned} $$ Since $y^T$ is $1 \times m$ and appears on the **left** of $A$, this is called the **left null space**. ## RREF and the Four Subspaces ### Example Matrix $$ A = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \\ 2 & 4 & 6 & 8 \end{bmatrix} $$ ### Row Echelon Form (REF) $$ \text{REF} = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 0 & -4 & -8 & -12 \\ 0 & 0 & 0 & 0 \end{bmatrix} $$ ### Reduced Row Echelon Form (RREF) **Step 1**: Normalize pivot rows $$ \begin{bmatrix} 1 & 2 & 3 & 4 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 0 & 0 \end{bmatrix} $$ **Step 2**: Eliminate above pivots $$ \text{RREF} = \begin{bmatrix} 1 & 0 & -1 & -2 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 0 & 0 \end{bmatrix} $$ ### Analysis - **Rank**: $r = 2$ (two pivot columns) - **Free variables**: $n - r = 4 - 2 = 2$ (columns 3 and 4) - **Dimension of $N(A)$**: $n - r = 2$ - **Dimension of $C(A)$**: $r = 2$ - **Dimension of $C(A^T)$**: $r = 2$ - **Dimension of $N(A^T)$**: $m - r = 3 - 2 = 1$ ::: {.callout-note} ## Key Observation RREF is the unique simplest form of a matrix obtained through row operations that preserve the solution space. ::: ## Visualization ![Four Fundamental Subspaces](https://github.com/ickma2311/foundations/raw/main/MIT18.06SC/lecture10/4_subspaces.png) ## Dimension Relationships **For an $m \times n$ matrix $A$ with rank $r$:** ### In $\mathbb{R}^n$: - $\dim(C(A^T)) + \dim(N(A)) = r + (n - r) = n$ - Row space and null space partition $\mathbb{R}^n$ ### In $\mathbb{R}^m$: - $\dim(C(A)) + \dim(N(A^T)) = r + (m - r) = m$ - Column space and left null space partition $\mathbb{R}^m$ ::: {.callout-important} ## Key Insight The four subspaces come in complementary pairs that completely partition their ambient spaces. ::: ## Summary **The four fundamental subspaces:** 1. **Column space** $C(A)$: where outputs $Ax$ live 2. **Null space** $N(A)$: inputs that map to zero 3. **Row space** $C(A^T)$: perpendicular complement to null space in $\mathbb{R}^n$ 4. **Left null space** $N(A^T)$: perpendicular complement to column space in $\mathbb{R}^m$ **Dimension formula**: - $\dim(C(A)) = \dim(C(A^T)) = r$ - $\dim(N(A)) = n - r$ - $\dim(N(A^T)) = m - r$ **Total check**: $r + (n-r) = n$ and $r + (m-r) = m$ --- *Source: MIT 18.06SC Linear Algebra, Lecture 10*

Subspace	Notation	Lives in	Dimension	Solves
Column Space	\(C(A)\)	\(\mathbb{R}^m\)	\(r\)	\(Ax = b\)
Null Space	\(N(A)\)	\(\mathbb{R}^n\)	\(n - r\)	\(Ax = \mathbf{0}\)
Row Space	\(C(A^T)\)	\(\mathbb{R}^n\)	\(r\)	\(A^T y = c\)
Left Null Space	\(N(A^T)\)	\(\mathbb{R}^m\)	\(m - r\)	\(A^T y = \mathbf{0}\)

MIT 18.06SC Lecture 10: Four Fundamental Subspaces

Overview

Column Space: \(C(A)\)

Null Space: \(N(A)\)

Row Space: \(C(A^T)\)

Left Null Space: \(N(A^T)\)

RREF and the Four Subspaces

Example Matrix

Row Echelon Form (REF)

Reduced Row Echelon Form (RREF)

Analysis

Visualization

Dimension Relationships

In \(\mathbb{R}^n\):

In \(\mathbb{R}^m\):

Summary