Earlier quoted context omitted.
> As vectors, obviously they have inverses, additive inverses. Since vectors don't have multiplication, there is no multiplicative inverse A vector is pretty much by definition also a matrix, and there is a standard way to multiply matrices. You can define several inverses of a vector that way, though you can't define a unique inverse. The standard inner product is of course also an exceptionally typical way to multi…
No, a vector is defined as an object that has certain properties, like addition and scalar multiplication. It's a very general, and abstract concept. There are vector spaces of functions, with infinite dimension, but there are also vector spaces with a finite number of elements. So only some vectors can even be written as 1xN matrices, if that is what you're referring to. But even if you write a vector that way, it d…
The set of elements defined by certain properties of their addition and of their multiplication with the elements belonging to a set of scalars is named "vector space" by some and "linear space" by others.
According to the etymology of the word vector, "linear space" would be more appropriate. You have used "vector" with the meaning "element of a linear space", and what you have said is correct, except that for any "vector" as an element of a linear space, considered as a column vector, there exists a corresponding row vector, even in the infinite-dimensional case.
"Vector" means translation of the space, and this is what "vector" meant when the word was introduced by Hamilton. While the set of translations is a linear space a.k.a. a vector space in the generalized sense, the set of translations, i.e. vectors in the strict sense, has additional properties due to the multiplication operations that must be defined for "vectors" in their strict sense (which are needed e.g. to determine the angles between translations and the distances).
"Vectors" as elements of linear spaces are a very general notion, which appears in many domains, and for all linear spaces, including for those infinite-dimensional, you can define matrices, i.e. linear functions, and matrix multiplication, i.e. composition of linear functions, and also the correspondence between a 1xN vector and a Nx1 vector, more correctly between a vector and an associated linear form. The latter also exists for the infinite-dimensional case, even if it is less likely to use names like row vectors and column vectors (though the names bra vectors and ket vectors are still in use for the infinite-dimensional case).
For the infinite-dimensional case the vectors and the matrices become functions of 1 or of 2 parameters and the sums from the formulas of matrix multiplication become integrals.
While for most computer applications, "vectors" refer just to elements of linear spaces, most "vectors" used in models of physical systems are vectors in the original sense of the word, where not only the vector addition and the product with scalars matter, but the products of vectors also have an essential role and their meaning can be best understood in the context of the complete geometric algebra theory.