All books I looked through write that the linear equation is the equation that has a from:
$$a_1x_1+a_2x_2+\dots+a_nx_n=b$$
But is this really the definition of a linear equation? I thought that the definition should be the form of linear mapping. Like,
$$f(x+y)=f(x)+f(y)$$ $$f(ax)=af(x)$$
What is the definition of a linear equation?
On
There are two different concepts here.
$1)$ A linear equation such as
$$ a_1x_1+a_2x_2+\dots+a_nx_n=b$$ where the variables appear in the first degree.
$2)$ A linear transformation which satisfies
$$f(x+y)=f(x)+f(y)$$
$$f(ax)=af(x)$$
For example $$2x+3y=10$$ is a linear equation.
On the other hand $$f(x) = 4x $$ is a linear function or linear transformation.
The second defines a linear function, not a linear equation. For a matrix $A$, the function $f(x)=Ax$ is linear. For $A=(a_1,\ldots ,a_n)$ and $x=(x_1,\ldots,x_n)^T$, $Ax=a_1x_1+\cdots +a_nx_n$.