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Would it be correct to define the function $y = 2x$ as the matrix transormation $ \begin{bmatrix} 1 & 0\\ 2 & 0 \end{bmatrix}$ on any two-dimensional vector $\begin{bmatrix} x \\ y \end{bmatrix}$ where $x, y \in \mathbb{R}$?

It feels weird to pass in a two-variable argument, but at the same time the output over the entire domain does seem to be $y = 2x$.

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The function $y = 2x$ is a linear transformation $\Bbb R\to \Bbb R$, and as such is represented by the $1\times 1$ matrix transformation $[2]$ on any one-dimensional vector $[x]$.

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    So what would what I wrote be? Is it meaningful at all? It does give the same result, no? – 2017-01-30
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    @jeremyradcliff What you wrote is a transformation that projects the entire plane onto the _graph_ of that function. – 2017-01-30
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    ah I see...so is the distinction (or one of them at least) that the function I wrote has no inverse function because we lose the plane's second dimension when applying the transformation and therefore can't map back to the original input? – 2017-01-30
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    Yeah, that's about it. – 2017-01-30