I read recently that every function $f : \mathbb{R}^m \to \mathbb{R}^n$ can be written as $f(\mathbf{x}) = \mathbf{Ax}$ where $\mathbf{A}$ is a matrix of constants.
On the one hand this is intuitive because the elements of $f(\mathbf{x})$ will be linear combinations of the elements of $\mathbf{x}$. On the other hand the one-dimensional case that students usually learn first as their first "linear" equation is a little different:
$$y = mx + b$$
Notice the presence of the intercept term $b$ which doesn't appear to have an analogy in $f(\mathbf{x}) = \mathbf{y} = \mathbf{Ax}$.
As far as I can tell, $g(\mathbf{x}) = \mathbf{Ax} + \mathbf{b}$ still preserves linearity, and can express transformations that $f(\mathbf{x}) = \mathbf{Ax}$ cannot (e.g. add 50 to every element of $\mathbf{x}$ even when $\mathbf{x}=\mathbf{0}$).
We could define a function $h$ that is equivalent to $g$ that extends $\mathbf{x}$ to have an extra $1$ appended on the end, call it $\mathbf{x}_{+1}$ and where $A$ has an extra column to act on it, e.g. $\begin{bmatrix}b \\ 0 \\ 0 \\ \vdots\end{bmatrix}$. Call the new matrix $\mathbf{C}$. Then we could say $h(\mathbf{x})=\mathbf{Cx}_{+1}$. But we're no longer purely multiplying by a constant matrix.
So:
- When we say $y = mx + b$ is a linear equation and we say $\mathbf{y} = \mathbf{Ax}$ is a linear transformation, do we mean the same thing?
- Is it correct to say $f(\mathbf{x}) = \mathbf{Ax}$ can represent all linear transformations $\mathbb{R}^m \to \mathbb{R}^n$, provided we are free to modify the elements of $\mathbf{A}$?
- Is $g(\mathbf{x}) = \mathbf{Ax} + \mathbf{b}$ not also a linear transformation?
- When people say $f(\mathbf{x})$ can represent all linear transformations do they really mean $h(\mathbf{x})$ or $g(\mathbf{x})$?
Best Answer
I think there is just a minor terminological confusion here. A linear transformation $f$ is required to satisfy $f(\mathbf{0}) = \mathbf{0}$ and is represented by matrix multiplication: for some constant matrix $\mathbf{A}$, $f(\mathbf{x}) = \mathbf{A}\mathbf{x}$ for all $\mathbf{x}$. If you compose $f$ with a translation along a constant vector $\mathbf{b}$, say, you get a transformation $g$ satisfying $g(\mathbf{x}) = \mathbf{A}\mathbf{x} +\mathbf{b}$ for all $\mathbf{x}$. Such a $g$ is called an affine transformation.