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So as it turns out, most equations of the form y = mx + b don't satisfy the requirements for linearity. Specifically, b must equal zero for the equation to be linear (that is, for it to meet the superposition and homogeneity criteria). Realizing this blew my mind, since I always assumed that linear equations and linearity went hand in hand.

Now, I've heard that when b is nonzero the equation is "incrementally linear". Is this a legitimate term to use, and if so, can someone link to a definition/explanation of the term in an authoritative website. Every website I've come across either mentions the term in passing or doesn't seem very credible.

Second, can you have a linear equation (again, an equation meeting the superposition and homogeneity criteria) with more than one input? How would you check for linearity in such a system? Do you simply check one input at a time, setting all other values to zero?

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"Incrementally linear" is something I had never heard, but it can do. "Affine" is a more widely used term. Those terminologies are usually referred to functions, not to equations. For equations you say non-homogeneous linear equation. –  Giuseppe Negro Sep 4 '12 at 20:53
    
"Linear" is being used in two different senses here: one sense is that the graph is a line, and the other sense is in the sense of a linear transformation (en.wikipedia.org/wiki/Linear_map). In the language of linear transformations, $x \mapsto mx + b$ is an affine transformation (en.wikipedia.org/wiki/Affine_transformation). –  Qiaochu Yuan Sep 4 '12 at 20:53

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Another term that is in more widespread use is "affine". An affine map is a linear map with a translation. In your one-dimensional example, the translation is a shift by $b$ units upward.

You can have linear equations with multiple inputs. This is the essence of the field of Linear Algebra: the use of matrices to map vectors/transform vector spaces.

In such a case, you could have something like

$$ \begin{pmatrix} y_1 \\ y_2 \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x_1 \\ x_2\end{pmatrix}.$$

An affine transform would just be

$$ \begin{pmatrix} y_1 \\ y_2 \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x_1 \\ x_2\end{pmatrix} + \begin{pmatrix} b_1 \\ b_2\end{pmatrix}.$$

However, as the link above shows, it is possible to represent an affine transform as a single matrix-vector multiplication. The resulting matrix is a simple linear map. Your one-dimensional case can then be written as

$$\begin{pmatrix} y \\ 1\end{pmatrix} = \begin{pmatrix} m & b \\ 0 & 1\end{pmatrix}\begin{pmatrix} x \\ 1\end{pmatrix}.$$

The matrix $A = \begin{pmatrix} m & b \\ 0 & 1\end{pmatrix}$ can be verified to be a linear map: Let $T :\Bbb R^n \to \Bbb R^n$ such that $T(x) = Ax$. Then $T(\alpha v + \beta y) = \alpha Ax + \beta Ay$, which we can verify using matrix-vector multiplication.

This property, which is a short-hand way of writing the properties of linearity, is how we check for linearity. In other words, $T(\alpha v + \beta y) = \alpha Ax + \beta Ay$ must hold for all $x, y \in \Bbb R^n$ for a linear map. That is to say, that a function is linear if it satisfies the following properties for all $x$:

  • f(ax) = af(x)
  • f(x+y) = f(x) + f(y)

When we want to look at multiple inputs, we turn $x$ into a vector, and the function becomes a transformation, or mapping. In linear cases, we can write this transformation as a matrix, which is, in some ways, a shorter way to write the function.

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