Whether multiplication by a constant matrix alone can represent every linear transformation

Question

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 $g(\mathbf{x})$?

Answer 1

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.

Answer 2

Just to sum up what others have said or alluded to: 'linear' has slightly different meanings in mathematics. It is well established terminology that a 'linear transformation' is one that can be represented by $\mathbf{x}\mapsto A\mathbf{x}$ where $A\mathbf{x}$ means ordinary multiplication of matrices. On the other hand $y=mx+c$ is the equation of a (non-vertical) line and so has some claim to be a 'linear' equation even though the map $x\mapsto mx+c$ is not a linear map (unless $c$ happens to be zero) as you rightly point out.

It is however noting that affine maps (in other words ones that map $\mathbf{x}\mapsto A\mathbf{x}+\mathbf{b}$) while not typically linear, can nevertheless be represented by matrices -- not by the usual matrix multiplication but by a slight modification of it. Write the input vector $\mathbf{x}$ with an extra bottom entry equal to 1. Then put $$\left(\begin{array}{c}\mathbf{y}\\ 1\end{array}\right)=\left(\begin{array}{cl}A & \mathbf{b}\\ \mathbf{0} & 1\end{array}\right)\left(\begin{array}{c}\mathbf{x}\\ 1\end{array}\right).$$

(Here $\mathbf{0}$ is a row vector of the appropriate length.) Then $\mathbf{y}$ is the image of $\mathbf{x}$ under the affine map $\mathbf{x}\mapsto A\mathbf{x}+\mathbf{b}$. Moreover the composition of affine maps corresponds to multiplication of matrices of this form.

Answer 3

In general, the property of being linear is defined as:

$f(x)$ is linear if $f(ax) = af(x)$ and $f(x+y)=f(x)+f(y)$. If $A$ is representing a linear transformation $T$, then the above tells us we need both of the following to be true:

$T(a{\bf x}) = aT({\bf x})$ and $T({\bf x} + {\bf y}) = T({\bf x}) + T({\bf y})$.

Now consider the two possibilities:

1. $T({\bf x}) = A{\bf x} + {\bf b}$
2. $T({\bf x}) = A{\bf x}$

Only the second satisfies the definition given above, so only the second is called linear.