Null Space ¶

The null space, denoted as $\text{Nul}A$ is the solution set of a homogeneous linear system, i.e. $Ax=0$.

A null space is a always a subspace of $\mathbb{R}^n$, why? Because one solution can always be the origin $(0, 0, ...)$.

As an example, consider a linear system.

$$ 2x_1-x_2+x_3 = 0\\ x_1+2x_2+3x_3= 0 $$

The augmented matrix is

$$ \left[ \begin{matrix} 2 & -1 & 1 & 0\\ 1 & 2 & 3 & 0 \end{matrix} \right] $$

Before solving the system, we have already known there is no unique solution since a free variable presents, due to the fact that two equation with three variables.

Solve for the reduced echelon form.

$x_3$ is a free variable, the solution set can be written as

$$ \left[ \begin{matrix} x_1 \\ x_2 \\ x_3 \end{matrix} \right]= \left[ \begin{matrix} -x_3 \\ -x_3 \\ x_3 \end{matrix} \right]= x_3\left[ \begin{matrix} -1 \\ -1 \\ 1 \end{matrix} \right] $$

which is a line passing both origin $(0, 0, 0)$ and $(-1, -1, 1)$, also a subspace of $\mathbb{R}^3$.

Now consider another example, suppose we have an augmented matrix

The solution can be written as:

$$ \left[ \begin{matrix} x_1 \\ x_2 \\ x_3 \\x_4 \\ x_5 \end{matrix} \right]= \left[ \begin{matrix} 2x_2+x_4-3x_5 \\ x_2 \\ -2x_4+2x_5 \\x_4 \\ x_5 \end{matrix} \right]= x_2\left[ \begin{matrix} 2 \\ 1 \\ 0 \\0 \\ 0 \end{matrix} \right] + x_4\left[ \begin{matrix} 1 \\ 0 \\ -2 \\1 \\ 0 \end{matrix} \right] +x_5\left[ \begin{matrix} -3 \\ 0 \\ 2 \\0 \\ 1 \end{matrix} \right] $$

The $\text{Nul}A$ is a subspace in $\mathbb{R}^5$ with $\text{dim}A=3$.

Null Space vs Col Space ¶

Consider matrix $A$

Column space is a subspace in $\mathbb{R}^n$, what is $n$? It is the number of rows, $n=3$.

Null space is a subspace in $\mathbb{R}^m$, what is $m$? It is the number of columns, $m=4$.

How to find any nonzero vector in $\text{Col}A$ and in $\text{Nul}A$?

Any column in a matrix can be a nonzero vector in $\text{Col}A$, for instance first column: $(2, -2, 3)^T$.

But to find a nonzero vector in null space requires some effort, construct the augmented matrix then turn it into rref.

The solution set with a free variable $x_3$ (because column 3 has no pivot) is

$$ \left[ \begin{matrix} x_1 \\ x_2 \\ x_3\\x_4 \end{matrix} \right]= \left[ \begin{matrix} -9x_3 \\ 5x_3 \\ x_3\\0 \end{matrix} \right] $$

If we pick $x_3 =1$, a nonzero vector in $\text{Nul}A$ is $(-9, 5, 1, 0)^T$

Now consider two vectors

$$ u = \left[ \begin{matrix} 3 \\ -2 \\ -1\\ 0 \end{matrix} \right],\qquad v = \left[ \begin{matrix} 3 \\ -1\\3 \end{matrix} \right]\\ $$

Is $u$ in $\text{Nul}A$? It can be verified easily

$Au\neq \mathbf{0}$, therefore $u$ is not in $\text{Nul}A$.

Is $v$ in $\text{Col}A$? Construct an augmented matrix with $v$, then solve it

The augmented matrix show there are solutions, i.e. $v$ is a linear combination of its column space basis, so $v$ is in $\text{Col}A$.

Row Space ¶

The Row space denoted as $\text{Row}A$, contains all linear combination of row vectors and subspace in $\mathbb{R}^n$.

If we perform row operations on $A$ to obtain $B$, both matrices have the same row space, because $B$'s rows are linear combinations of $A$'s. However, row operation will change the row dependence.

An Example ¶

Find the row, column and null space of

The basis of the row space of $B$ is its first 3 rows: $(1,0,1,0,1), (0, 1, -2, 0, 3), (0, 0, 0, 1, -5)$ which are also the basis of the row space of $A$. However it does not necessarily mean that first 3 rows of $A$ forms the basis for row space, because the dependence among rows changed by row operation.

In constrast, the basis of col space of $A$ is $(-2, 1, 3, 1)^T, (-5, 3, 11, 7)^T, (0, 1, 7, 5)^T$.

The null space is

$$ \left[ \begin{matrix} x_1 \\ x_2 \\ x_3\\x_4 \\x_5 \end{matrix} \right]= \left[ \begin{matrix} -x_3-x_5 \\ 2x_3-3x_5 \\ x_3\\5x_5 \\x_5 \end{matrix} \right]= x_3\left[ \begin{matrix} -1 \\ 2 \\ 1\\0 \\0 \end{matrix} \right]+ x_5 \left[ \begin{matrix} -1 \\ -3 \\ 0\\5 \\1 \end{matrix} \right] $$

Rank ¶

Definition of rank: The rank is the dimension of the column space of $A$. The nullity of $A$ is the dimension of the null space.

The Rank Theorem¶

The dimensions of the column space and the row space of an $m \times n$ matrix $A$ are equal. Therefore, the rank is the same for either column space and row space.

This common dimension, the rank of $A$, also equals the number of pivot positions in $A$ and satisfies the equation: $$ \operatorname{rank} A + \operatorname{dim} \mathrm{Nul} A = n $$

The intuition behind this is that when a matrix $A$ is converted into its reduced row echelon form (rref) $B$, we can indirectly determine the basis of the column space by matching the columns of $B$ with those of $A$. The columns in $A$ that correspond to pivot columns in $B$ form the basis of the column space.

In the rref, we can also directly see the basis of the row space. Each row in the basis of the row space must contain a pivot. The rows that do not contain pivots correspond to the free variables, which determine the dimension of the null space.

Example 1 ¶

If $A$ is $45 \times 50$ matrix with a $10$-dimension nullity, what is the rank of $A$?

$10$-$D$ nullity means 10 free variables, so the pivots are $50-10=40$, which is also the rank of $A$.

Example 2 ¶

The matrices below are row equivalent. $$ A=\left[\begin{array}{rrrrr} 2 & -1 & 1 & -6 & 8 \\ 1 & -2 & -4 & 3 & -2 \\ -7 & 8 & 10 & 3 & -10 \\ 4 & -5 & -7 & 0 & 4 \end{array}\right], \quad B=\left[\begin{array}{rrrrr} 1 & -2 & -4 & 3 & -2 \\ 0 & 3 & 9 & -12 & 12 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] $$

1. Find rank $A$ and $\operatorname{dim}$ Nul $A$
2. Find bases for Col $A$ and Row $A$.
3. What is the next step to perform to find a basis for Nul $A$ ?
4. How many pivot columns are in a row echelon form of $A^{T} ?$

1. $rank(A)=2$, because $B$ has two pivots. And nullity is the number of free variables, there are 3, so $\text{dim Nul}A = 3$.

2. Bases for $\text{Col}A$ is $(2,1,-7,4)^T, (-1,-2,8,-5)^T$, and for $\text{Row}A$ is $(1,-2,-4,3,-2),(0,3,9,-12,12)$.

The $\text{Nul}A$ and basis is

$$ \left[ \begin{matrix} x_1 \\ x_2 \\ x_3\\x_4 \\x_5 \end{matrix} \right]= \left[ \begin{matrix} -2x_3+5x_4-6x_5 \\ -3x_3+4x_4-4x_5 \\ x_3\\x_4 \\x_5 \end{matrix} \right]= x_3 \left[ \begin{matrix} -2 \\ -3 \\ 1\\0 \\0 \end{matrix} \right]+ x_4 \left[ \begin{matrix} 5 \\ 4 \\ 0\\1 \\0 \end{matrix} \right]+ x_5 \left[ \begin{matrix} -6 \\ -4 \\ 0\\0 \\1 \end{matrix} \right] $$

3. Perform rref on augmented $A$

4. Transpose $A$ then do rref.

There are 2 pivot columns.

Actually, we don't need any calculation to know the rank of $A^T$, because

$$ rank(A)=rank(A^T) $$

Orthogonality of $\text{Nul}A$ and $\text{Row}A$ ¶

$\text{Nul}A \perp \text{Row}A$ ¶

Here is the intersting connections of these subspaces we have discussed. Consider

The basis of row space of $A$ is $(1, 0, 0)$ and $(0, 1, .25)$.And the $\text{Row}A$ is

$$ \text{Row}A= s\left[ \begin{matrix} 1 \\ 0\\ 0 \end{matrix} \right]+ t\left[ \begin{matrix} 0 \\ 1\\ 0.25 \end{matrix} \right] $$

The $\text{Nul}A$ is $$ \left[ \begin{matrix} x_1 \\ x_2\\ x_3 \end{matrix} \right]= x_3 \left[ \begin{matrix} 0 \\ -.25\\ 1 \end{matrix} \right] $$

Now we can visualize their relations geometrically. Again keep in mind that Matplotlib does not render 3D properly, so you need some imagination as well.

Here is what we observe.

The $\text{Row}A$ is a plane and $\text{Nul}A$ is a line which is perpendicular to the plane (maybe 3D plot not so obvious about it). It is easy to grasp the idea if you notice that in a homogeneous system $Ab = \mathbf{0}$, it breaks down into dot products

$$ Ab =\left[ \begin{matrix} A_{1\cdot}\cdot b \\ A_{2\cdot}\cdot b\\ A_{3\cdot}\cdot b \end{matrix} \right] = \left[ \begin{matrix} 0 \\ 0 \\ 0 \end{matrix} \right] $$

where $A_{1\cdot}, A_{2\cdot}, A_{3\cdot}$ are the rows of $A$. In later chapters we will prove when the dot product of two vectors equals zero, they are geometrically perpendicular.

Let me write a summary:

1. The dimension of the row space is equal to the dimension of the column space (both are the rank of the matrix).
2. The row space of $A$ is orthogonal to the null space of $A$.
3. The column space of $A$ is orthogonal to the left null space of $A$.

$\text{Nul}A^T \perp \text{Col}A$ ¶

The nullity of $A^T$ is

The $\text{Nul}A^T$ is

$$ \left[ \begin{matrix} x_1 \\ x_2\\ x_3 \end{matrix} \right]= x_2 \left[ \begin{matrix} -2 \\ 1\\ 0 \end{matrix} \right] $$

The $\text{Col}A$ is

$$ \text{Col}A= s\left[ \begin{matrix} 5 \\ 10\\ 3 \end{matrix} \right]+ t\left[ \begin{matrix} 8 \\ 16\\ 4 \end{matrix} \right] $$

$\text{Col}A$ is a plane and $\text{Nul}A^T$ is a line perpendicular to the plane. The intuition is similar to $\text{Nul}A \perp \text{Row}A$, here you can think of a system look like $b^TA = \mathbf{0}^T$.

Rank Decomposition ¶

Consider a matrix $A$, the purpose is to decompose it into the multiplication of $C$, $R$, which are the bases of column space and row space respectively.

$$ A = CR $$

Get the basis of $\text{Col}A$.

Then get the $\text{Row}A$.

Multiply $CR$, we are getting back $A$.

Verify if $CR$ equals $A$.