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A friend of mine was criticized in undergrad by a Professor for saying that a matrix is a function.

Now, a matrix can be represented by a linear transformation, and linear transformations by definition are functions.

Is there any theoretical reason as to why a matrix can't be dubbed a function? From my understanding, they are functions. Am I missing something?

user90275
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  • "A matrix can be represented by a linear transformation." What does this even mean? – Michael Albanese Oct 08 '13 at 03:14
  • @MichaelAlbanese "..we develop a one-to-one correspondence between matrices and linear transformations that allows us to utilize properties of one to study properties of the other." - Friedberg, Linear Algebra. The chapter name is "The Matrix Representation of a Linear Transformation". That is what I mean by a matrix can be represented by a linear transformation. – user90275 Oct 08 '13 at 03:21
  • See my answer below. You may not have covered this yet but that one-to-one correspondence relies on a choice of bases on $\mathbb{R}^n$ and $\mathbb{R}^m$, but I'm guessing you are (implicitly) taking the standard bases on these spaces. – Michael Albanese Oct 08 '13 at 03:23

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A linear transformation is a function.

Given a linear transformation $T : \mathbb{R}^n \to \mathbb{R}^m$, and a choice of bases on $\mathbb{R}^n$ and $\mathbb{R}^m$, there is an $m\times n$ matrix $A$ such that $T(x) = Ax$ for all $x \in \mathbb{R}^n$. Note, choices were made to write $T$ in this form. Note, $A$ is not the function, it is just used in the definition of the function.

In the other direction, given an $n\times m$ matrix $A$, and a choice of bases on $\mathbb{R}^n$ and $\mathbb{R}^m$, you can define a linear transformation $T : \mathbb{R}^n \to \mathbb{R}^m$ by $T(x) = Ax$ but again, $A$ is not the function, $T$ is.

Consider the analogous situation of a function $f : \mathbb{R} \to \mathbb{R}$ given by $f(x) = 2x$. Would you say that $2$ is a function? Here $2$ plays the role of the matrix $A$. This isn't so much an analogy as it is the special case $n = m = 1$.

Michael Albanese
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  • What you have said is true. However, a linear transformation has a matrix representation (i.e. a one-to-one correspondence between linear transformations and matrices.) More formally, the matrix representation of $T: \mathbb{R}^n \rightarrow \mathbb{R}^m$ in the ordered based $\beta$ and $\gamma$ is $A = [T]^{\gamma}_{\beta}$ and $A \in \mathbb{R}^{m \times n}$ – user90275 Oct 08 '13 at 03:38
  • How is that different from what I said? – Michael Albanese Oct 08 '13 at 03:39
  • Matrix $A$ is a linear transformation. A linear transformation is a function. I don't understand why $A$ cannot be a function. – user90275 Oct 08 '13 at 03:43
  • The matrix $A$ is not a linear transformation. The one-to-one correspondence is a way of pairing a matrix with a linear transformation, it does not say the two things are equal. For example, there is one-to-one correspondence between ${1, \dots, 26}$ and ${a, \dots, z}$, that doesn't mean every letter is a number. – Michael Albanese Oct 08 '13 at 03:46