Null and Column Spaces

We've seen one important subspace of $\large \mathbb{R}^n$, the span of a set of vectors. There are two more subspaces that play a big role in the applications of linear algebra, and we'll take a look at both of those here.

Null Space

Definition: Let $\large A$ be an $\large m\times n$ matrix. The null space of $\large A$, or $\large \text{Null}(A)$, is the set of all the solutions of $\large A\vec{x}=\vec{0}$.

Example: Define the matrix $\large A$ and vector $\large \vec{v}$:

$$\large A = \begin{bmatrix} 1 & -3 & -2 \\ -5 & 9 & 1 \\ \end{bmatrix}, \vec{v} = \begin{bmatrix} 5 \\ 3 \\ -2 \\ \end{bmatrix} $$

Is $\large \vec{v}\in\text{Null}(A)$? Simple matrix-vector multiplication shows that $\large A\vec{v}$ does equal $\large \vec{0}$, so yes, $\large \vec{v}\in\text{Null}(A)$.

Theorem: If $\large A$ is an $\large m\times n$ matrix, then $\large \text{Null}(A)$ is a subspace of $\large \mathbb{R}^n$.

Note

You will prove this theorem as part of HW. But note that if $\large A$ is $\large m\times n$, then $\large \vec{x}$ must be $\large n\times 1$, so vectors in $\large \text{Null}(A)$ are in $\large \mathbb{R}^n$.

Let's ask a few questions to check our understanding of this subspace:

• True or False: $\large \vec{0}$ is always in $\large \text{Null}(A)$?

Click to see the answer

 

True! Two reasons: we can see directly that $\large A\vec{0}=\vec{0}$, but we also know that $\large \text{Null}(A)$ is a subspace of $\large \mathbb{R}^n$, and every subspace of $\large \mathbb{R}^n$ contains $\large \vec{0}$.

 

• If we know $\large \vec{0}$ is the only vector in $\large \text{Null}(A)$, what do we know about solutions of the linear system $\large A\vec{x}=\vec{0}$?

Click to see the answer

 

Then $\large A\vec{x}=\vec{0}$ has only one solution, $\large \vec{x}=\vec{0}$.

 

How do we describe $\large \text{Null}(A)$? Let's see an example. Define $\large A$ as:

$$\large A = \begin{bmatrix} -3 & 6 & -1 & 1 & -7 \\ 1 & -2 & 2 & 3 & -1\\ 2 & -4 & 5 & 8 & 4 \\ \end{bmatrix} $$

Find a set of vectors that spans $\large \text{Null}(A)$. The video shows how we approach this type of problem:

Column Space

Definition: Let $\large A$ be an $\large m\times n$ matrix. The column space of $\large A$, or $\large \text{Col}(A)$, is the set of all possible linear combinations of the columns of $\large A$. In other words, if the columns of $\large A$ are $\large \vec{a_1}, \vec{a_2}, \dots, \vec{a_n}$, then $\large \text{Col}(A) = \text{Span}\lbrace \vec{a_1}, \vec{a_2}, \dots, \vec{a_n}\rbrace$.

Here's a helpful way to view $\large \text{Col}(A)$: it is the set of every $\large \vec{b}$ such that $\large A\vec{x}=\vec{b}$ for some $\large \vec{x}$. It's every vector that can be reached through an $\large A\vec{x}$ product. Here's one more way to say it: $\large \vec{b}$ is in the column space of $\large A$ if $\large A\vec{x}=\vec{b}$ is consistent.

Theorem: If $\large A$ is an $\large m\times n$ matrix, then $\large \text{Col}(A)$ is a subspace of $\large \mathbb{R}^m$.

Note

You will prove this theorem as part of HW.

Think through this...why is $\large \text{Col}(A)$ a subspace of $\large \mathbb{R}^m$ instead of $\large \mathbb{R}^n$?

Click to see the answer

 

The column space is the span of the column vectors, and if $\large A$ is $\large m\times n$, then the columns of $\large A$ are vectors of length $\large m$.

 

Here's an example of the type of question you can expect from this section: Define $\large A$ and $\large \vec{b}$ as follows:

$$\large A = \begin{bmatrix} 2 & 1 & 0 \\ -1 & 3 & 5 \\ \end{bmatrix}, \vec{b} = \begin{bmatrix} 1\\ -6\\ \end{bmatrix} $$

Is $\large \vec{b}$ in the column space of $\large A$? The video works out the solution to this problem:

Note

We can define the row space of $\large A$ as the set of all linear combinations of the rows of $\large A$. The definition works fine, but this is a less useful space than the column space, and we won't do much with it.

A "Good" Square Matrix

With these subspaces, we can now add to our list of equivalencies for square matrices:

The following are all equivalent facts about a square $\large n\times n$ matrix $\large A$.

1. $\large A$ is invertible.
2. $\large \text{det}(A)\ne 0$.
3. $\large A$ is row equivalent to $\large I$, the identity matrix.
4. $\large A\vec{x}=\vec{0}$ has only the trivial solution.
5. $\large A\vec{x}=\vec{b}$ has a unique solution for any $\large \vec{b}$.
6. $\large \text{Col}(A)=\mathbb{R}^n$.
7. $\large \text{Null}(A)=\lbrace \vec{0}\rbrace$ (the set containing only the zero vector).

Definition: A square matrix $\large A$ that meets the above criteria is called nonsingular. Otherwise, the matrix is called singular.

Section Outcomes

I can:

• Recall and use the definitions of the Null Space and Column Space of a matrix.
• Determine if a given vector is in $\large \text{Null}(A)$ or $\large \text{Col}(A)$.
• Find a set of vectors in $\large \text{Null}(A)$.