I want to know whether the following series of thought is valid.
My textbook says an equation of the form $a_1 x_1 + a_2 x_2 + …+ a_n x_n = b$ where $a_1 , … , a_n $ and $ b$ is some constant is linear equation.
According to the textbook, the following equations are linear
$x_1 + 5x_2 – 2 x_3 =1 $, $x_1 = -7x_2 +3x_3$, $3x+28y-1/2z=0$
and the followings are nonlinear
$x_1 + 3x_2 + x_1 x_3 =2 $ , $xyz=x+2y$, $x^{-2} + x_2 +8x_3 =5$
I think the definition textbook gave is useful for judging linearness of nonlinearness of the equations at one sight, but it doesn't click my mind because it does not match with my intuition with linearity.
I think the story should go like this… 'these are the $\textbf{algebraic differences}$ between linear and nonlinear equation and from these, we can prove that an equation should has a form $a_1 x_1 + a_2 x_2 + …+ a_n x_n = b$ to be linear…'
So I thought a while to figure it out, and found some algebraic differences between linear and nonlinear equation. which is I think crucial to do 'elimination'.
1. If I double (or multiply constant)the variables and constant, the equation remain balanced.
e.g. from the above examples, $(2x_1)+5(2x_2)-2(2x_3)=2(1)$ but $(2x)(2y)(2z)=8xyz\neq2x+4y=(2x)+2(2y)$.
2. If you add two equations, it still holds property 1.
So the question is : how can I demonstrate that an equation which has the property 1 and 2 (which I think is linear) should has the form $a_1 x_1 + a_2 x_2 + …+ a_n x_n = b$ ? Thanks for reading.
Best Answer
Do you have some background in linear algebra? A map $f:\mathbb{R}^n \to \mathbb{R}$ is linear if $f(x+y) = f(x) + f(y)$ and $f(\alpha x) = \alpha f(x)$ for $x,y \in \mathbb{R}^n$, $\alpha \in \mathbb{R}$. I believe these are the properties you are looking for. It is easy to see (*) that any linear map $f:\mathbb{R}^n \to \mathbb{R}$ has the form $f(x_1,\ldots,x_n) = \alpha_1 x_1 + \cdots + \alpha_n x_n$ for some scalars $\alpha_i$. Thus your textbook states that a linear equation is one of the form $f(x_1,\ldots,x_n) = b$ for some linear map $f:\mathbb{R}^n \to \mathbb{R}$.
(*) If $f:\mathbb{R}^n \to \mathbb{R}$ is linear, define $\alpha_i := f(e_i)$ where $e_i = (0,\ldots,0,1,0,\ldots,0)$ ($1$ in the $i$th position) is the $i$th standard basis vector. Then $$f(x_1,\ldots,x_n) = f(x_1 e_1 + \cdots + x_n e_n) = x_1 f(e_1) + \cdots + x_n f(e_n) = \alpha_1 x_1 + \cdots + \alpha_n x_n.$$