ennuvolat4gv

2022-02-16

The linear equation is the equation that has a from:
${a}_{1}{x}_{1}+{a}_{2}{x}_{2}+\dots +{a}_{n}{x}_{n}=b$
But is this really the definition of a linear equation? I thought that the definition should be the form of linear mapping. Like,
$f\left(x+y\right)=f\left(x\right)+f\left(y\right)$
$f\left(ax\right)=af\left(x\right)$
What is the definition of a linear equation?

Miles Martin

The second defines a linear function, not a linear equation. For a matrix A, the function f(x)=Ax is linear. For
$A=\left({a}_{1},\dots ,{a}_{n}\right)$ and $x={\left({x}_{1},\dots ,{x}_{n}\right)}^{T}$,
$Ax={a}_{1}{x}_{1}+{a}_{2}{x}_{2}+\dots +{a}_{n}{x}_{n}$

Andrew Fenton

There are two different concepts here.
1) A linear equation such as ${a}_{1}{x}_{1}+{a}_{2}{x}_{2}+\dots +{a}_{n}{x}_{n}=b$ where the variables appear in the first degree.
2) A linear transformation which satisfies $f\left(x+y\right)=f\left(x\right)+f\left(y\right)$
$f\left(ax\right)=af\left(x\right)$
For example $2x+3y=10$ is a linear equation.
On the other hand $f\left(x\right)=4x$ is a linear function or linear transformation.

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