Why is there implied an equality between vectors and $n$-tuples?

Question

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? :)

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

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.

Answer 5

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