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I was explaining something to my friend when we ran into this question that I couldn't clearly answer.

Does $(a,b)=\begin{bmatrix}a\\b\end{bmatrix}$? Are these different notations for a single object sitting in one particular space, or are these distinctive objects in different spaces?

We say that $(a,b)\in \mathbb{R}^2$, but we also say $\begin{bmatrix}a\\b\end{bmatrix}\in\mathbb{R}^2$. No big deal...but then do we say $\begin{bmatrix}a&b\end{bmatrix}\in\mathbb{R}^2$ also?

I have only learned Cartesian product, $$\mathbb{R}\times\mathbb{R}=\{(a,b)\mid a,b\in\mathbb{R}\}.$$ Is $$\mathbb{R}\times\mathbb{R}=\left\{\begin{bmatrix}a\\b\end{bmatrix}\mid a,b\in\mathbb{R}\right\}$$ another common definition? Are these even distinct or just two ways of naming the set?

MattW
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  • It depends how you define $\mathbb{R}^2$..... trivially. – Dan Rust Nov 02 '17 at 00:56
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    I think when specifying a particular matrix shape, it's more correct (and certainly more consistent with the convention for general rectangular matrices) to write $\mathbb R^{2 \times 1}$ for the set of column vectors and $\mathbb R^{1 \times 2}$ for the set of row vectors, and reserve $\mathbb R^{2}$ to refer to ordered pairs $(a,b)$ which are not matrices at all. But that's just my opinion. Certainly there are authors who disagree, so it's best to be flexible and infer the notation from the context. –  Nov 02 '17 at 00:57
  • One serious unavoidable question: what you mean by "the same thing". Also, what is the definition of ${\bf R}^2$? –  Nov 02 '17 at 01:11
  • I will try to clarify. – MattW Nov 02 '17 at 01:17
  • It's a common convention (though certainly not universal) to identify column vectors (in $\mathbb R^{2 \times 1}$) with points (in $\mathbb R^2$), while treating the row vector $[a ; b]$ as a different object obtained by transposing; i.e. $[a; b] = (a,b)^T.$ – Anthony Carapetis Nov 02 '17 at 01:52
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    No matter how you look at it, theses usually aren't vectors in $\mathbb{R}^2$. ;-) (That is, there's a typo in the title.) – Hans Lundmark Nov 02 '17 at 05:39

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