📂Matrix Algebra

Row Space, Column Space, Null Space

Definition¹

$$A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix}$$

For a matrix $A$, the $m$ number of $\mathbb{R}^{n}$ vectors made from the rows of $A$

$$\begin{align*} \mathbf{r}_{1} =& \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \end{bmatrix} \\ \mathbf{r}_{2} =& \begin{bmatrix} a_{21} & a_{22} & \cdots & a_{2n} \end{bmatrix} \\ &\vdots \\ \mathbf{r}_{m} =& \begin{bmatrix} a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix} \end{align*}$$

are called the row vectors of $A$. The $n$ number of $\mathbb{R}^{m}$ vectors made from the columns of $A$

$$\mathbf{c}_{1} = \begin{bmatrix} a_{11} \\ a_{21} \\ \vdots \\ a_{m1} \end{bmatrix},\quad \mathbf{c}_{2} = \begin{bmatrix} a_{12} \\ a_{22} \\ \vdots \\ a_{m2} \end{bmatrix},\quad \dots,\quad \mathbf{c}_{n} = \begin{bmatrix} a_{1n} \\ a_{2n} \\ \vdots \\ a_{mn} \end{bmatrix}$$

are called the column vectors of $A$.

$$\begin{align*} A =& \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix} = \begin{bmatrix} \mathbf{r}_{1} \\ \mathbf{r}_{2} \\ \vdots \\ \mathbf{r}_{m} \end{bmatrix} \\ =& \begin{bmatrix} \mathbf{c}_{1} & \mathbf{c}_{2} & \cdots & \mathbf{c}_{n} \end{bmatrix} \end{align*}$$

The $\mathbb{R}^{n}$ subspace generated by the row vectors $\mathbf{r}_{1}, \mathbf{r}_{2},\dots,\mathbf{r}_{m}$ of $A$ is called the row space of $A$ and is denoted as follows.

$$\mathcal{R} (A) \quad \text{or} \quad \text{row}(A)$$

The $\mathbb{R}^{m}$ subspace generated by the column vectors $\mathbf{c}_{1}, \mathbf{c}_{2},\dots,\mathbf{c}_{n}$ of $A$ is called the column space of $A$ and is denoted as follows.

$$\mathcal{C} (A) \quad \text{or} \quad \text{col}(A)$$

The set of solutions to the homogeneous system of linear equations $A \mathbf{x} =\mathbf{0}$ is called the null space of $A$ and is denoted as follows.

$$\mathcal{N}(A) \quad \text{or} \quad \text{null}(A)$$

Explanation

Such concepts were devised to solve

$$\begin{equation} A\mathbf{x} = \mathbf{b} \end{equation}$$

systems of linear equations. That is, in linear algebra, there is interest in the relationship between the solutions of $(1)$ and the row space, column space, and null space of $A$. In particular, finding the basis of the row space is related to solving linear systems. Specifically, the dimension of the row space and column space is called the rank, and the dimension of the null space is called the nullity.

Note that the column space is also called the image. If one considers the matrix $A$ as a concept of a function, then the function corresponding to $A \in \mathbb{R}^{m \times n}$ can also be seen as $T_{A} : \mathbb{R}^{n} \to \mathbb{R}^{m}$.

Theorem 1

A necessary and sufficient condition for the linear system $A \mathbf{x} = \mathbf{b}$ to have a solution is $\mathbf{b} \in \mathcal{C}(A)$.

Theorem 2

Let $\mathbf{x}_{0}$ be some solution to $A\mathbf{x} = \mathbf{b}$. Let us call $S= \left\{ \mathbf{v}_{1}, \mathbf{v}_{2}, \dots, \mathbf{v}_{k} \right\}$ a basis for $\mathcal{N}(A)$. Then, all solutions of $A\mathbf{x} = \mathbf{b}$ can be expressed as follows.

$$\begin{equation} \mathbf{x} = \mathbf{x}_{0} + c_{1}\mathbf{v}_{1} + c_{2}\mathbf{v}_{2} + \cdots + c_{k}\mathbf{v}_{k} \end{equation}$$

Conversely, for all constants $c_{1}, c_{2}, \dots, c_{k}$, the above $\mathbf{x}$ is a solution to $A\mathbf{x} = \mathbf{b}$.

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$(2)$ is called the general solution of $A \mathbf{x} = \mathbf{b}$. $\mathbf{x}_{0}$ is called the particular solution of $A \mathbf{x} = \mathbf{b}$. Furthermore, $c_{1}\mathbf{v}_{1} + c_{2}\mathbf{v}_{2} + \cdots + c_{k}\mathbf{v}_{k}$ is called the general solution of $A \mathbf{x} = \mathbf{0}$.

From these theorems, it can be understood that the general solution of a nonhomogeneous linear system can be represented as the sum of a particular solution and the general solution of the homogeneous linear system.

See Also

Kernel in Abstract Algebra

The null space is written as $\ker A$ and also called the kernel. This is because it is a specialization of the general concept of kernel discussed in abstract algebra, seeing $A$ as a function.