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Are they considered equal in some sense?

For instance, I always write "...for vector $\mathbf{x} \in {\mathbb{R}}^n$ we have ...". I have a small problem with this (not a big one). The problem comes from the fact that the "cartesian power" of a set is defined as: ${\mathbb{R}}^n = \underbrace{ \mathbb{R} \times \mathbb{R} \times \cdots \times \mathbb{R} }_{n}= \{ (x_1,\ldots,x_n) \mid x_i \in \mathbb{R} \ \text{for all} \ 1 \le i \le n \}$. In other words, we are implying that the vector $\mathbf{x}$ is a tuple because we, with the relation "is an element of" ($\in$), are defining the $n$-dimensional vector to be a member of a set ${\mathbb{R}}^n$ in where members are $n$-tuples.

Is there implied a tiny mathematical abuse of notation here? Or have I completely lost it? :)

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    @niyazi: But I thought that we gain most power from seeing vectors as constructs that are exactly _not_ attached to a point?2011-12-05

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The word "vector" is used to mean different things in different contexts. A point in $\mathbb{R}^n$ is often called a "vector".

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    But a point in $\mathbb{R}^n$ _is_ a vector since $\mathbb{R}^n$ is a vector-space - right?2011-12-06
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I don't understand the problem. A vector is an element of a vector space. A vector space is any set equipped with appropriate operations satisfying the vector space axioms, and $\mathbb{R}^n$ is an example of such a thing.

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    No there is probably not :)2011-12-05
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In general, a vector is not a tuple. But in the specific case of $\mathbb{R}^n$, where $n$ is a natural number, the elements of this space are tuples, because that is how the space is defined.

Nevertheless, although these vectors happen to be tuples, it is often more elegant to pretend that they are atomic objects, and work in a "coordinate free" way. This emphasizes the geometry of the vector space rather than its algebraic aspects.

On the other hand, given a finite dimensional vector space $V$ over a field $F$, say of dimension $k$, the original space is isomorphic to the vector space $F^k$ of $k$-tuples of elements of $F$. So in a sense these "tuple spaces" capture all finite dimensional vector spaces up to isomorphism.

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    But they all these spaces are isomorphic to $\mathbb{R}^n$ of $k$-tuples of elements as you said so it can be said that they are tuples? And something else: What is the difference when you consider n-tuple as a $m \times 1$ or $1 \times n$ matrix? AFAIK tuples are different structures from matrices. Can you explain this please?2015-05-16
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Once you choose a basis for your vector space, you get a one-to-one correspondence with $\mathbb R^n$. For most purposes, then, we may say that the vector space is $\mathbb R^n$. But not for all purposes. At a certain point in their education, a math student has to learn to think of an abstract vector space not given as a space of $n$-tuples.

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    Thanks. It seems that I had missed a few points with the more fundamental definitions. Se my comment on Qiaochu's answer. If you a are working enterily in $n$-dimensional euclidean space over the field $\mathbb{R}$ (or $\mathbb{C}$) it seems to be a fully valid assumption that ${\mathbb{R}}^n$ is in fact a _vector space_. hey, I never studied this ... just fooling around with wikipedia and some books :)2011-12-05
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You can kind of say that a vector is a tuple but there isn't much to gain by doing that. When working with vectors we usually care more about the linear algebraic properties (vectors can be summed, mutiplied by scalars, etc) and less concerned about combinatorial and structural properties (like saying things are a tuple or not).

Also, the tuple analogy falls off a bit when you move to infinite-dimensional vector spaces, like functions or polynomials.

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    Well, computer proof systems, like Coq, also usually give every value a type, and values of different types are usually not interchangeable. It's just that in mathematics, converting between tuples and vectors is often a natural thing to do (so natural that we don't even bother saying we're doing it), just like in computing, converting between, say, characters and numbers is often a natural thing to do (although since our audience is a computer, not a human, we have to explicitly say so). In short, math and programming are not that different in this regard.2011-12-05