Linear and nonlinear systems

Question

I get confused about the difference between linear and nonlinear system. Suppose that we have a linear system

\begin{equation}\label{1} Ax=b \tag{1} \end{equation}

with $A \in \mathbb{R}^{n\times n}$ and $b\in \mathbb{R}^{n}$and we define the $x$ as a nonlinear function

\begin{equation} x(t) = \sin(2\pi t) + \sin(3\pi t) \quad \text{where} \quad t=1,2,\dots,n \tag{2} \end{equation}

my question is, does the linearity of the system (1) depends on the nature of $x$ ( that can be either linear or nonlinear function)?. In case that the system (1) stays always linear as long as it takes the form of $A x$ no matter how the $x$ is defined, is it possible to transform it to a nonlinear system by adding a nonlinear term ? A term like $e^{x(t)}$ or $x(t)^{2}$ ? is the system $Ax(t) + e^{x(t)} = b$ a nonlinear one or example?

Thank you for you explanations and help.

Answer 1

In a linear system of equation, $x$ is the vector of variables. It is not a vector of functions, so speaking about $x(t)$ is not really sensible. In your writing, you first say that $x\in\mathbb R^n$, but then you use the term $x(t)$. This is nonsensical. $x$ can either be a vector, or it can be a function, but it cannot be both at the same time.

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That said, "a linear system" is an ill-defined term. There are multiple mathematical concepts that are called "a linear system etc". For example, there are linear systems of differential equations but you probably aren't talking about those.

A linear system of equations is any set of linear equations. Any such set of equation can be written compactly as $Ax=b$ for some fixed matrix $A$ and some fixed vector $b$.

Answer 2

Let us first consider a system of equations, where we have only one equation. Then it is called linear if it is of the form $$ a_1x_1+\cdots +a_nx_n=b, $$ for constants $a_i,b\in K$ and variables $x_i$. It is called polynomial, if it is of the form $$ f(x_1,\ldots ,x_n)=0, $$ for a polynomial $f\in K[x_1,\ldots ,x_n]$. For example, $$ 3z^2y^7-28xyz+17z-28xy+515=0 $$ in the variables $x,y,z$ and rational coefficients.

It is non-linear if it is not linear. Now a system of equations just means that we have several such equations, which we want to solve simultaneously.

For example, consider the following system of polynomial, non-linear equations $$x+y+z=1$$ $$x^2+y^2+z^2=35$$ $$x^3+y^3+z^3=97$$