2. Systems of linear equations and matrices
For example,
$$ \begin{cases} 5 x_1 + 2 x_2 = 4\\ 3 x_1 + 6 x_2 = 7 \end{cases} $$
is a system of two linear equations, both of then involving two variables $x_1$ and $x_2$ (the unknowns). The numbers $\ 5,\ 2$ and $\ 3,\ 6$ are called the coefficients of the system, and $\ 4$ and $\ 7$ are the constant terms.
A general system of $m$ linear equations with $n$ unknowns can be written as
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$ \begin{align*} \left\{ \begin{aligned} a_{11}x_1 +\, a_{12}x_2\, +\, \dots\, +\, a_{1n}x_n&=b_1\\ a_{21}x_1 +\, a_{22}x_2\, +\, \dots\, +\, a_{2n}x_n&=b_1\\ \dots\qquad\qquad\qquad\qquad&\\ a_{m1}x_1 + a_{m2}x_2 + \dots + a_{mn}x_n&=b_m\\ \end{aligned} \right. \end{align*}, $ |
where $\ x_1,\ x_2,\ \dots\ ,\ x_n$ are the unknowns, $\ a_{11},\ a_{12},\ \dots\ ,\ a_{mn}$ are the coefficients of the system, and $\ b_1,\ b_2,\ \dots\ ,\ b_m $ are the constant terms. According to matrix multiplication
$$ \begin{pmatrix} a_{11}x_1 +\, a_{12}x_2\, +\, \dots\, +\, a_{1n}x_n\\ a_{21}x_1 +\, a_{22}x_2\, +\, \dots\, +\, a_{2n}x_n\\ \vdots\\ a_{m1}x_1 + a_{m2}x_2 + \dots + a_{mn}x_n \end{pmatrix}= \begin{pmatrix} a_{11}&a_{12}&\dots &a_{1n}\\ a_{21}&a_{22}&\dots &a_{2n}\\ \vdots & \vdots &\ddots&\vdots\\ a_{m1}&a_{m2}&\dots &a_{mn} \end{pmatrix} \begin{pmatrix} x_1\\ x_2\\ \vdots\\ x_m \end{pmatrix}. $$
Since two matrices are equal if and only if all the corresponding elements of the two matrices are equal, and we have
$$ a_{11}x_1 +\, a_{12}x_2\, +\, \dots\, +\, a_{1n}x_n=b_1,\\ a_{21}x_1 +\, a_{22}x_2\, +\, \dots\, +\, a_{2n}x_n=b_2,\\ \dots\\ a_{m1}x_1 + a_{m2}x_2 + \dots + a_{mn}x_n=b_m, $$
we can write down the system $(\ast)$ in matrix notation as follows
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$$ \begin{pmatrix} a_{11}&a_{12}&\dots &a_{1n}\\ a_{21}&a_{22}&\dots &a_{2n}\\ \vdots & \vdots &\ddots&\vdots\\ a_{m1}&a_{m2}&\dots &a_{mn} \end{pmatrix} \begin{pmatrix} x_1\\ x_2\\ \vdots\\ x_m \end{pmatrix}= \begin{pmatrix} b_1\\ b_2\\ \vdots\\ b_m \end{pmatrix}. $$ |
What is the matrix notation for
Also, the augmented matrix
$$ \left(\begin{array}{rrrr|r} a_{11}&a_{12}&\dots &a_{1n}&b_1\\ a_{21}&a_{22}&\dots &a_{2n}&b_2\\ \vdots & \vdots &\ddots&\vdots&\vdots\\ a_{m1}&a_{m2}&\dots &a_{mn}&b_m \end{array}\right), $$
where the firts row represence the first equation, the second row represence the second equation and so on, has exactly the same information as the original system $(\ast\ast)$. The augmented matrix can be used for solving the system of linear equations.
For example, let us consider solving the system
$ \begin{align*} \left\{ \begin{aligned} 1x - 2y &= 2\\ 2x + 4y &= 12 \end{aligned} \right. \end{align*} $
by using the facts that
- we can multiply an equation (that is, every term of an equation) with a nonzero constant without changing the solution, e.g.,
$5x+3=-2\quad \Longleftrightarrow \quad\color{blue}{\frac{1}{5}}\cdot5x+\color{blue}{\frac{1}{5}}\cdot3=\color{blue}{\frac{1}{5}}\cdot(-2)$;
- an equation which is obtained by adding some value to both sides of an initial equation, has the same solution as the initial equation, e.g.
$5x+3=-2\quad \Longleftrightarrow \quad 5x+3\color{blue}{-3}=-2\color{blue}{-3}$,
but also $\quad 2x+y=7\quad \Longleftrightarrow \quad 2x+y\color{blue}{+4x-y}=7\color{blue}{-2}\text{,}\quad$ if $\quad 4x-y=-2$.
Thus, a step-by-step solution for the system is
$ \begin{align*} \left\{ \begin{array}{rrrrr} 1x & - & 2y & = & 2\\ 2x & + & 4y & = & 12 \end{array} \right. \overset{R_2-2\cdot R_1}\Longleftrightarrow \left\{ \begin{array}{rrrrr} 1x & - & 2y & = & 2\\ & + & 8y & = & 8\\ \end{array} \right. \overset{\frac{1}{8}\cdot R_2}\Longleftrightarrow \left\{ \begin{array}{rrrrr} 1x & - & 2y & = & 2\\ & & y & = & 1\\ \end{array} \right. \overset{R_1+2\cdot R_2}\Longleftrightarrow \left\{ \begin{array}{rrrrr} x & & & = & 4\\ & & y & = & 1\\ \end{array} \right. \end{align*}, $
where $R_1$ denotes the first equation and $R_2$ denotes the second equation. It can be written down also in terms of matrices:
$ \begin{align*} \left(\begin{array}{rr|r} 1 &-2 &2\\ 2 & 4 &12\\ \end{array}\right) \quad\overset{R_2-2\cdot R_1}\longrightarrow\quad \left(\begin{array}{rr|r} 1 &-2 &2\\ 0 & 8 &8\\ \end{array}\right) \quad\overset{\frac{1}{8}\cdot R_2}\longrightarrow\quad \left(\begin{array}{rr|r} 1 &-2 &2\\ 0 & 1 &1\\ \end{array}\right) \quad\overset{R_1+2\cdot R_2}\longrightarrow\quad \left(\begin{array}{rr|r} 1 & 0 &4\\ 0 & 1 &1\\ \end{array}\right)\enspace, \end{align*} $
where the first row of the matrices re-presence the first equation and the second row of the matrices re-presence the second equation.
Thus, solving a system of linear equation can be viewed as manipulating the corresponding augmented matrix rows. It is something a computer can easily do and thus, it makes possible to solve a system of linear equations by a computer.
The algorithm of making matrix row manipulations for solving the system of linear equations is known as Gaussian elimination algorithm.