0
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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
1
$\begingroup$

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 $x$".

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.