# What is the difference between an n-tuple and a n-dim column vector or an n-dim row vector?

I've recently started studying linear algebra and I've been confused by the way that in most written texts, n-tuples and n-dim column or row vectors (or sometimes even matrices) seem to be used interchangeably. I understand that an n-tuple could in some sense be thought of as a similar mathematical object to a (1 x n) matrix (and I guess even an (n x 1) matrix); however, when it comes to vectors I'm completely lost. One of the greatest sources of confusion for me is that:

We're not really specifying a basis for our column vector (or row vector); and does this mean that we're assuming that the column vector under consideration has been expressed under the standard unit vectors when we equate it to an n-tuple?

I'd be grateful if you'd help me understand this; thank you.

• Vectors in a vector space are generally written as column vectors while covectors in a dual space can be thought of as row vectors. – JG123 Nov 23 '19 at 18:48
• For your second question, I would think that if you are given an arbitrary column vector without any other specification, it is assumed that it is an element of $\mathbb{R^n}$ (which is also a vector space, so the column vector would be a linear combination of the standard basis vectors). Of course, the column vector could be an element of some other vector space V, where it would be a linear combination of the basis vectors of V. – JG123 Nov 23 '19 at 19:00

An $$n$$-tuple $$x$$ is a function $$x:\quad[n]\to{\rm world},\qquad k\mapsto x_k\quad(1\leq k\leq n)\ .$$ The data inherent in such an $$n$$-tuple is completely represented by the list $$(x_1,x_2,\ldots, x_n)$$. E.g., (Meyer, Hans, 1978, German, Palo Alto) could be a $$5$$-tuple.
In mathematics $$n$$-tuples are mostly of the form $$x:\quad[n]\to X,\qquad k\mapsto x_k\in X\quad(1\leq k\leq n)\ ,$$ where $$X$$ is a ground set specified in advance, e.g., $$X={\mathbb R}$$. The $$n$$-tuples $$x=(x_1,\ldots,x_n)$$ are then nothing else but elements of $$X^n$$. Note that so far the $$n$$-tuples are just lists, or arrays, of (mathematical) objects.
Now in linear algebra such $$n$$-tuples are used for various purposes. In particular they occur as $$(1\times n)$$-matrices $$[x_1 \ x_2\ \ldots\ x_n]$$ or as $$(n\times 1)$$-matrices $$\left[\matrix{x_1\cr x_2\cr\vdots\cr x_n\cr}\right]\ ,\tag{1}$$ depending on their momentaneous use. Hereby the following agreement is very common: The $$n$$-tuples $$(x_1,\ldots, x_n)$$ denoting the coordinates of a point $$x\in {\mathbb R}^n$$ are tacitly interpreted as column vectors $$(1)$$. In this way the equation $$y=Ax$$ can be read as "$$y$$ is the image point of $$x$$ under the linear map $$A$$", and "the column vector $$y$$ is the product of the matrix $$A$$ with the column vector $$x$$".