Things that matter

We now introduce a more compact way of writing a system of linear equations. Consider the following system: \begin{eqnarray} u - 2v + w & = & 4\\ -u -2w & = & 5 \end{eqnarray} where $$u, v,$$ and $$w$$ are the variables.

First of all, this system can be written in tuple form as $$\begin{bmatrix} u- 2v + w\\ -u - 2w\end{bmatrix} = \begin{bmatrix} 4\\5\end{bmatrix}$$. (Note that the equal sign is applied component-wise.)

We now introduce a new way of writing the system as the matrix equation: $$\begin{bmatrix} 1 & -2 & 1 \\ -1 & 0 & -2 \end{bmatrix} \begin{bmatrix} u \\ v \\ w \end{bmatrix} = \begin{bmatrix} 4 \\ 5 \end{bmatrix}.$$

There are three items in the representation of a system of linear equations as a matrix equation. Starting from the left, we have a matrix of the coefficients called the coefficient matrix. The first row of the matrix is formed from the coefficients in the first equation. The second row is formed from the coefficients in the second equation.

The order in which the coefficients appear is important. One needs to fix in advance which column of the matrix represents which variable. The second item from the left is a tuple of the variables listed in the correct order. The first entry of this tuple, $$u$$ in this case, is the variable that corresponds to the first column of the coefficient matrix. The second entry, $$v$$, corresponds to the second column of the coefficient matrix. The third entry, $$w$$, corresponds to the third column of the coefficient matrix. Hence, the system of linear equations could also be represented as $$\begin{bmatrix} 1 & 1 & -2 \\ -2 & -1 & 0 \end{bmatrix} \begin{bmatrix} w \\ u \\ v \end{bmatrix} = \begin{bmatrix} 4 \\ 5 \end{bmatrix}.$$ Finally, the item of the right is a tuple consisting of the right-hand side values.

To indicate an evaluation of the left-hand side at a particular assignment of values, say, with $$u = 1$$, $$v = 2$$, and $$w = 3$$, we write $$\begin{bmatrix} 1 & -2 & 1 \\ -1 & 0 & -2 \end{bmatrix} \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$$ which evaluates to $$\begin{bmatrix} 1\cdot 1 + (-2)\cdot 2 + 1\cdot 3 \\ (-1)\cdot 1 + 0\cdot 2 + (-2)\cdot 3 \end{bmatrix} = \begin{bmatrix} 0 \\ -7 \end{bmatrix}.$$

If $$A$$ denotes the coefficient matrix, $$x$$ denotes the tuple of variables, and $$b$$ denotes the tuple of the right-hand side values, we can write the system compactly as $Ax = b.$

General case

Let $$\mathbb{F}$$ be a field. Let $$m$$ and $$n$$ be positive integers. Let $$A \in \mathbb{F}^{m \times n}$$, $$x = \begin{bmatrix} x_1 \\ \vdots \\ x_n\end{bmatrix}$$, and $$b = \begin{bmatrix} b_1 \\ \vdots \\ b_m\end{bmatrix}$$ where $$x_1,\ldots,x_n$$ are variables and $$b_1,\ldots,b_m\in \mathbb{F}$$. (Here, $$\mathbb{F}^{m \times n}$$ denotes the set of matrices with $$m$$ rows and $$n$$ columns whose entries are elements of $$\mathbb{F}$$. So $$A$$ is such a matrix.)

We define $$Ax$$ to be the $$m$$-tuple given by $\begin{bmatrix} A_{11}x_1 + A_{12}x_2 + \cdots + A_{1n}x_n \\ A_{21}x_1 + A_{22}x_2 + \cdots + A_{2n}x_n \\ \vdots \\ A_{m1}x_1 + A_{m2}x_2 + \cdots + A_{mn}x_n \end{bmatrix}$ where $$A_{ij}$$ denote the entry in row $$i$$ and column $$j$$ of $$A$$, called the $$(i,j)$$-entry of $$A$$.

Hence, $$Ax = b$$ represents the system of linear equations given by $\begin{array}{c} A_{11}x_1 + A_{12}x_2 + \cdots + A_{1n}x_n = b_1 \\ A_{21}x_1 + A_{22}x_2 + \cdots + A_{2n}x_n = b_2 \\ \vdots \\ A_{m1}x_1 + A_{m2}x_2 + \cdots + A_{mn}x_n = b_m \end{array}$ The system has $$m$$ equations and $$n$$ unknowns.

Exercises

1. Let $$A = \begin{bmatrix} 3 & 1 \\ 4 & 5 \end{bmatrix}$$. What is the $$(2,1)$$-entry of $$A$$?

2. For each of the following, write out the system of linear equations represented by $$Ax = b$$.

1. $$A = \begin{bmatrix} 1 & 0 & -2 \\ 0 & -3 & 4 \end{bmatrix}$$, $$x = \begin{bmatrix} x \\ y \\ z \end{bmatrix}$$, and $$b = \begin{bmatrix} 5 \\ -6 \end{bmatrix}$$.

2. $$A = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0\end{bmatrix}$$, $$x = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$$, and $$b = \begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix}$$.