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.

  • $\begingroup$ Vectors in a vector space are generally written as column vectors while covectors in a dual space can be thought of as row vectors. $\endgroup$ – JG123 Nov 23 '19 at 18:48
  • $\begingroup$ 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. $\endgroup$ – 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 $x$".


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