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.