# Linearity of the function $f(x) = ax + b$

According to Wikipedia (1)(2), linear function and linear equation are different. Further more, the function $f(x) = ax + b$ (where $a$ and $b$ are real numbers and $ab \neq 0$) is not a linear function but an affine function when $b \neq 0$, while the equation $y = ax + b$ is still called a linear equation.

But in the internet, I do not see any document states like that. In another word, they think that linear function and linear equation are the same (except that equations are more general than functions).

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For a "linear function" we want $f(\lambda x)=\lambda f(x)$, which isn't true for $f(x)=ax+b$ unless $b=0$. – Mark Bennet Oct 20 '12 at 18:28
In a very crude sense, a linear function is equivalent to multiplying a vector with a matrix. That's it, nothing more. In case you have $y = ax+b$, you can think of a to be a 1x1 matrix and x to be 1x1 vector but the $b$ causes problems. This prevents the function from being called a linear function. – Inquest Oct 20 '12 at 18:30
$ax+b$ could for example also be a linear polynomial or occur as linear factor of a polynomial and the graph of this function is a line (whence the name "linear" in the first place). Unfortuanately, the adjective "linear" comes in differnt flavours. It would be more consequent and less ambiguous to always speak of affine linear if $b$ could be nonzero. – Hagen von Eitzen Oct 20 '12 at 18:31

Of course a linear function is not the same thing as a linear equation. But I imagine that the question intends to ask what is meant by a linear function, explicitly in the one-variable case.

That unfortunately is context-dependent. Usage differs. While I would never call $2x-1$ a linear function of $x$ while teaching linear algebra, I have often called $2x-1$ the linear approximation to the function $x^2$ in a neighbourhood of $x=1$.

I suspect it is standard practice in high school to call $2x-1$ a linear function of $x$. In the formal sense, is a linear function of $x-\frac{1}{2}$.

And not only in high school. For example, see this. A glance at Google hits for the search "linear function" shows a number of pictures of straight lines that do not pass through the origin.

In statistics, a linear (predictor) function is a function of shape $a_0+a_1x_1+a_2x_2+\cdots+a_n x_n$. So allowing a non-zero constant term is not only a one-variable phenomenon.

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what do you mean by stating that "In the formal sense, is a linear function of x−1/2"? Sorry for my being bad at Math. – Hoang Royal Oct 20 '12 at 21:18
@HoangRoyal: Well, $2x-1=2(x-1/2)$. So if we make the change of variable $u=x-1/2$, we are looking at the function $2u$. – André Nicolas Oct 20 '12 at 21:28

The difference is that one talks about linear functions usually between vector spaces. Since in vector spaces you have a scalar multiplication and addition, you would like for your functions between these structures to respect these same properties, then a linear function between two vector spaces must satisfy $f(x+y) = f(x) + f(y)$ and $f(\lambda x) = \lambda f(x)$ for any two vectors $x,y$ in the domain.

In the other case you're talking about polynomial equations, which are classified by their degree. A linear equation is a polynomial equation of degree one; it has this name because the graph of a polynomial of degree one in $n$ variables is a line in an $n$ dimensional Euclidean space.

Then why the "abuse of notation"? Well, because vector spaces try to generalize the geometric notion of "flatness", by which I mean that if you want to represent an $n$ dimensional vector space, this won't have bendings to it (think of a line, or a plane). So, in some sense, they are linear.

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The word 'linear' is used for a few different but related things.

A linear map (or linear function) is a function between vector spaces (over the same field, say $F$), say $f : V \to W$, which satisfies $f(\lambda v) = \lambda f(v)$ for all $\lambda \in F$ and $v \in V$. So for instance, the linear maps $\mathbb{R} \to \mathbb{R}$ are precisely the functions of the form $f(x) = \lambda x$.

A linear polynomial (in one variable) is a polynomial of the form $p(x)=ax+b$.

A linear equation (in a particular set of variables) is an equation where each individual term is either constant or is a linear function of one of the variables. So if our variables are $x,y,z$, say, then $ax+by+cz=d$ is linear but $xy=1$ and $ax^2+bx+c=0$ are not.

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A collection of equations is called linear if it is the sum of terms that are either constant or the product of a variable and a constant. The reason we call such equations "linear" is that their solutions (if they exist) are lines or their generalizations to higher dimensions, like planes, etc.

However, the solution to a linear equation may not describe a linear map, because the graph of a linear map needs to intersect the origin. So not every line you can draw actually describes a linear map, but it does describe the solution to a linear equation.

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To add to what others say, the equation of a line in two dimensions is $y=mx+c$ - as I was taught it. As an operator this is a stretch (by $m$) and a translation (by $c$). Although translations are important (e.g. in dynamics - the equations of motion), and seem obviously to be linear in character, it turns out that it is very useful to have a concept of linearity which applies when the origin is fixed - i.e. we ignore translations, or they are unimportant in the context we are thinking about.

This idea turns out to be useful beyond the immediate world of vector spaces (tensor products, bilinear and quadratic forms, modules over a ring, linear differential equations, representation theory etc).

Basically a function $f$ is said to be linear if there is a system of "scalars" around (like a field or a ring), and if $\lambda$ and $\mu$ are arbitrary scalar factors then $f(\lambda a + \mu b) = \lambda f(a) + \mu f(b)$. That would be a basic definition to go with your "linear algebra" tag.

As other answers have suggested, there are other uses of the word "linear" in mathematics - there have been several questions on this site asking for clarification - it isn't always clear, and you just have to get used to it. The word affine is sometimes used to make it clear that translations are included.

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