What exactly do we call a linear equation?

Question

I know there's a difference between the linear algebra "linear function" and the "affine function", but it confused me about what we consider a "linear equation" -precisely how many variables allowed, what order could a variable carry?

Answer 1

Here are various comments on linear functions versus linear equations.

First there is the idea of a linear combination. Essentially, linear combination means a "sum of multiples" of something. For example

$$3x-2y+4z$$

is a linear combination of the variables $x,y$ and $z$, that is, a sum of multiples of $x,y$ and $z$.

If you set this expression equal to a constant value such as

$$3x-2y+4z=7$$

then you have a linear equation.

A linear function, on the other hand, is a function with the property that the function of a linear combination of two or more variables equals the same linear combination of the function of the two variables. This definition is difficult to understand without an example.

Let $f(x)=3x$. Then find $f(4a+7b)$

Solution:

$$f(4a+7b)=3(4a+7b)=12a+21b$$

Now look at the same linear combination of $f(a)$ and $f(b)$.

$$4f(a)+7f(b)=4(3a)+7(3b)=12a+21b$$

It is the same. That is

$$f(4a+7b)=4f(a)+7f(b)$$

This notion of linearity occurs in many areas of mathematics.

Notice, for example, that definite integrals are linear. For example

$$\int_0^15x-6x^2\,dx=5\int_0^1x\,dx-6\int_0^1x^2\,dx$$

Differentiation is also linear. For example

$$ \frac{d}{dx}\left(3\sin x+5e^x\right)=3\frac{d}{dx}\sin x+5\frac{d}{dx}e^x $$

Should, at some point, you study Laplace transforms you will find that they are linear transforms.

$$ \mathcal{L}\{a\cdot f(t)+b\cdot g(t)\}=a\cdot\mathcal{L}\{f(t)\}+b\cdot\mathcal{L}\{g(t)\} $$