Null Space and Column Space of a Matrix

Hello! Welcome back.

In our last lesson, we saw that a matrix representing a linear transformation is fundamentally tied to our choice of basis. By changing our perspective (the basis), we change the matrix, even though the underlying geometric action remains the same. This led us to the powerful idea of finding a special basis, an "eigenbasis," where the transformation's matrix becomes simple and diagonal.

Today, we're going to dig deeper into the fundamental geometry of any linear transformation $T (x) = A x$ . We will analyze the transformation by asking two critical questions:

What is the set of all possible outputs? This set, called the column space or image, tells us where the transformation can actually send vectors.
Which vectors get "squashed" down to the zero vector? This set, the null space or kernel, reveals what information is lost by the transformation.

Our goal is to learn how to compute the column space and null space of a matrix. Understanding these two subspaces is essential for analyzing everything from the behavior of quantum operators to the flow of information through neural network layers.

1. The Column Space: The Range of the Transformation

Let's start with the first question: what are the possible outputs of a transformation $T (x) = A x$ ? The set of all possible output vectors is called the column space of the matrix $A$ , denoted as $Col (A)$ . It's also known as the image of the transformation.

Geometrically, if a transformation squishes 3D space onto a plane, that plane is its column space. If it collapses space onto a line, that line is its column space.

The Null Space & Column Space of a Matrix | Algebraically & Geometrically

To build some intuition, let's watch a short clip from Dr. Trefor Bazett that introduces the column space geometrically as the set of all 'targets' of a transformation.

Watch from the beginning to 1:13.

So, how do we find this set of all possible outputs? Remember that the matrix-vector product $A x$ is just a linear combination of the columns of $A$ , where the components of $x$ are the weights:

A x = x_{1} (col 1) + x_{2} (col 2) + \dots + x_{n} (col n)

This means the set of all possible outputs is precisely the span of the columns of A. This is why it's called the "column space"!

Definition: Column Space
The column space of an $m \times n$ matrix $A$ , denoted $Col (A)$ , is the span of the columns of $A$ . It is a subspace of the codomain, $R^{m}$ .

So, describing the column space is easy: it's the span of the column vectors. However, this set of vectors is often linearly dependent and thus contains redundant information. Our real task is to find a basis for the column space—a minimal set of vectors that still spans the entire space.

Finding a Basis for the Column Space

Here is a reliable method to find a basis for $Col (A)$ :

Row reduce the matrix $A$ to its reduced row echelon form (RREF).
Identify the columns in the RREF that contain a pivot (a leading 1).
The basis for the column space of $A$ is the set of the corresponding columns from the original matrix A.

This is a critical point: row operations can change the column space itself, but they preserve the linear dependence relationships among the columns. If the 3rd column of the RREF is a linear combination of the first two, then the 3rd column of the original matrix $A$ is also the same linear combination of its first two columns. Therefore, we use the RREF to identify which original columns are independent, and those original columns form our basis.

Null space and column space basis | Vectors and spaces | Linear Algebra | Khan Academy

The Khan Academy video 'Null space and column space basis' provides a detailed walkthrough of this process. We'll watch it in parts. First, let's see how Sal finds a basis for the column space.

First, watch from 0:11 to 1:35 to see the initial setup. Then, jump to 12:16 and watch until 23:45. In this second part, Sal has already computed the RREF and uses it to identify which of the original column vectors are redundant, leaving him with a basis for the column space. Pay close attention to how he shows that the non-pivot columns can be written as combinations of the pivot columns.

2. The Null Space: The Kernel of the Transformation

Now for our second question: which input vectors are mapped to the zero vector?

Geometrically, if a 2D transformation squishes the plane onto a line, there must be a whole line of vectors in the input space that gets collapsed onto the origin. This set of vectors is the null space.

Definition: Null Space
The null space of an $m \times n$ matrix $A$ , denoted $Nul (A)$ , is the set of all vectors $x$ in the domain $R^{n}$ that satisfy the homogeneous equation $A x = 0$ . It is also called the kernel of the transformation.

Null Space Definition and Example — This image provides the formal definition of the null space as the set of all solutions $\mathbf{x}$ to the equation $A\mathbf{x} = \mathbf{0}$, alongside a concrete example.

The null space is interesting because if it contains more than just the zero vector, the transformation is not one-to-one; multiple different input vectors are mapped to the same output vector.

Finding a Basis for the Null Space

Compared to the column space, finding a basis for the null space is more direct. The definition tells us exactly what to do: we need to find all solutions to $A x = 0$ .

Set up the augmented matrix $[A ∣ 0]$ and row reduce it to RREF.
Write out the corresponding system of linear equations.
Identify the pivot variables and free variables. The free variables correspond to the columns without pivots.
Express each pivot variable in terms of the free variables.
Write the general solution vector $x$ as a linear combination of vectors, with the free variables as the parameters. These vectors form a basis for the null space.

Null space and column space basis | Vectors and spaces | Linear Algebra | Khan Academy

Let's return to the Khan Academy video, this time to see the null space calculation for the same matrix. This part walks through the entire procedure, from row reduction to finding the basis vectors.

Watch from 1:35 to 12:16. Follow Sal's step-by-step process: (1) row reducing the matrix, (2) identifying pivot and free variables, and (3) expressing the solution in parametric vector form to get the basis for the null space.

Test your understanding!

Let's find the null space and column space for the matrix $A = [12 - 2 - 4 36]$ .

Find a basis for the column space of A.
Find a basis for the null space of A.

Show answer

First, we row reduce $A$ :

[12 - 2 - 4 36] R_{2} - 2 R_{1} [10 - 2 0 30]

This is the RREF of A.

1. Column Space Basis:

We identify the pivot columns in the RREF. There is only one pivot, in the first column.
Therefore, the basis for the column space of $A$ consists of the first column of the original matrix A.
Basis for $Col (A)$ : ${[12]}$ . The column space is a line in $R^{2}$ .

2. Null Space Basis:

We solve $A x = 0$ using the RREF. The system is $x_{1} - 2 x_{2} + 3 x_{3} = 0$ .
The pivot variable is $x_{1}$ . The free variables are $x_{2}$ and $x_{3}$ .
We express the pivot variable in terms of the free ones: $x_{1} = 2 x_{2} - 3 x_{3}$ .
Now, we write the general solution vector $x$ :

x = x_{1} x_{2} x_{3} = 2 x_{2} - 3 x_{3} x_{2} x_{3} = x_{2} 210 + x_{3} - 3 01

The vectors in this parametric form make up the basis for the null space.
Basis for $Nul (A)$ : $⎩ ⎨ ⎧ 210, - 3 01 ⎭ ⎬ ⎫$ . The null space is a plane in $R^{3}$ .

3. Summary: Two Fundamental Subspaces

The null space and column space describe the fundamental properties of a matrix transformation. It's crucial to remember where they "live" and how they are defined.

	Null Space ( $Nul (A)$ )	Column Space ( $Col (A)$ )
Definition	Set of all $x$ such that $A x = 0$ .	The span of the columns of $A$ .
Where it lives	A subspace of the domain ( $R^{n}$ ).	A subspace of the codomain ( $R^{m}$ ).
What it tells us	Which vectors get squashed to zero. If non-trivial, $A$ is not 1-to-1.	The range of possible outputs.
How to find a basis	Solve $A x = 0$ and write in parametric vector form.	Find pivot columns of $A$ 's RREF; use the original columns of $A$ .

The Null Space & Column Space of a Matrix | Algebraically & Geometrically

Dr. Trefor Bazett provides a concise summary comparing these two spaces. This is a great way to consolidate what we've learned.

Watch from 9:07 to 10:28 to review the key differences between the null space and the column space.

Conclusion

In this lesson, we dissected the action of a matrix transformation by defining and computing its two most important subspaces.

Key Takeaways:

The column space (or image) is the set of all possible outputs of a transformation. It's the span of the matrix's columns, and its basis is found using the pivot columns of the original matrix.
The null space (or kernel) is the set of all input vectors that are mapped to the zero vector. Its basis is found by finding the parametric vector solution to the homogeneous equation $A x = 0$ .
The column space is a subspace of the codomain ( $R^{m}$ ), while the null space is a subspace of the domain ( $R^{n}$ ).

Preview of the Next Lesson:
You might have noticed an interesting relationship in our exercise: the matrix was $2 \times 3$ . The dimension of the null space (number of free variables = 2) plus the dimension of the column space (number of pivot columns = 1) equaled 3, the total number of columns (the dimension of the domain). This is not a coincidence! In our next lesson, we will formalize this beautiful and powerful relationship with the Rank-Nullity Theorem.