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This may sound like an obvious question but it has confused me! According to wikipedia (https://en.m.wikipedia.org/wiki/Vector_(mathematics_and_physics)) vectors are defined as:

"An element of a vector space"

But can't you have a vector space with elements of matrices, and for that matter numbers or even functions. Does this not mean that all of these are vectors to (along with the normal arrow like vectors)?

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    $\begingroup$ Yes, being a "vector" is not really an intrinsic property of the object, but a property that states that it "lives" in some vector space, and as such, there is some way it can be added to the other elements of that space and multiplied by scalars from the ground field. $\endgroup$ Sep 4, 2014 at 7:50
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    $\begingroup$ This is an example where the same word is used in different ways. In some contexts "vector" means specifically an ordered $n$-tuple of real numbers. In some contexts "vector" means an equivalence class of directed line segments (with respect to a certain equivalence relation). Sometimes "vector" just means any element of a particular vector space. Which definition of "vector" is being used must be determined from context. $\endgroup$
    – littleO
    Sep 4, 2014 at 8:02
  • $\begingroup$ Generally a vector is a kind of $n\times 1$ (or $1\times n$, I allways forget the convention) matrix. With the same arithmetic rules. (Multipling, Substracting, ...) But I have never heart that someone names a matrix a vector. $\endgroup$
    – Matthias
    Sep 4, 2014 at 8:09

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The term vector can have two different meanings.
A. Most people particularly physics students & learn vectors as direction and magnitude. This meaning has an alternate representation/ viewpoint as coordinates in space (2 dimensions, 3 dimensions or higher).
In the introductory linear algebra courses I am familiar with, most examples of vectors are of this first type, so fit well as a column inside a matrix.

B. From a linear algebra & higher math perspective, the term vector has a much broader, more abstract meaning. First define a vector space, then as you mentioned a vector is an element in that space.

For example, you can have a vector space of functions, such as the vector space consisting of all polynomials of degree $3$ or less (including the zero polynomial). Addition and scalar multiplication are defined in the standard way.

Another example is the space of all continuous real valued functions defined on an interval $[a,b]$. (We can define an inner product on this space by $\langle f, g \rangle = \int_a^b f g \, dx$.)

Some clarification as to where you're coming from would be helpful to better tailor the answer to your subject. Physics, Calculus/Math of 2 or 3 dimensions, linear algebra, etc.

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Yes. If $K$ is the set of $n\times n$ matrices with elements in a field $F$, then you can regard $K$ as a vector space over $F$. The multiplication would be multiplying every element of a matrix by the same element $\lambda\in F$. So technically you could regard the elements of $K$ as vectors.

However, matrices are rarely referred to as vectors because of the potential confusion. In particular, matrices are frequently used to represent linear transformations between vector spaces - if $A$ is a matrix in $K$ and $v$ is an ordinary $n\times 1$ column vector with elements in $F$, then $Av$ is another vector (using ordinary matrix multiplication.

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  • $\begingroup$ What do you mean this is never done? It is done all the time, for example when we regard the set of $n\times n$ matrices as an algebra over the ground field. $\endgroup$ Sep 4, 2014 at 8:59
  • $\begingroup$ Are you saying it is common to refer to matrices as vectors in that context? I have not encountered that. Or are you saying that I have expressed myself badly - probably true! I meant that matrices are not referred to as vectors, but rereading it that is not what I said. I will edit. Better, or still wrong? $\endgroup$
    – almagest
    Sep 4, 2014 at 9:20

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