# What does vector mean in Linear Algebra?

I have just started reading Linear Algebra and there are some basic things I cannot understand. I read some answers on this site and also tried to search in some books but I didn't find a clear answer. Here are few words form my textbook : 1. Here by VECTOR we do not mean the vector quantity which we have defined in vector algebra as a directed line segment. 2. Matrices having a single row or column are referred to as vectors. 3. I also watched a video in which at approximately 3:55 he says that a point in two dimensional real coordinate space is written in matrix form in LINEAR ALGEBRA.

Now I'm confused! Since in the video it seems as a vector in LINEAR ALGEBRA is just same as a point in 1, 2, 3...n real coordinate spaces. But in my text book its written that vector is not the vector quantity!

1. And I also read in a book in which it was written that we are using LINEAR word instead of VECTOR to avoid confusion!

So is it really the difference in words? Why its not written clearly what the vector really is.

• A vector is an element of a vector space. A space is a set with some property attaced. The term "vector" is letting you know which properties are attached. They include, amongst other things, multiplication by a scalar. Many things can be seen as vectors including functions. – Karl Dec 25 '15 at 18:39
• No @janmarqz I do not think its clear enough for ME to understand! – shivani Dec 25 '15 at 18:43
• The word "vector" means different things in different contexts. Sometimes "vector" means an ordered $n$-tuple of numbers. Sometimes "vector" means an equivalence class of directed line segments. Sometimes "vector" simply means "an element of a vector space" that we are working with. Hopefully context will make it clear which meaning of the word "vector" is being used. – littleO Dec 25 '15 at 18:53
• I suppose that one can think of a vector in linear algebra as a coordinate vector. Given an ordered basis of a vector space $V$ over a field $\Bbb{K}$, one can express every element of $V$ as a possibly infinite row matrix (or column matrix, depending on your preference) with finitely many non-zero entries in $\Bbb{K}$. – Transcendental Dec 26 '15 at 7:52

## 5 Answers

A vector by definition is an element of some vector space. Unless specified otherwise, this is the definition you should have in mind. Now let me try to clear up some of your specific questions.

1. Here by VECTOR we do not mean the vector quantity which we have defined in vector algebra as a directed line segment.

Without context, it's impossible for me to figure out exactly what is being meant here. Most likely, they previously introduced a specific vector space, such as $\mathbf{R}^n$ and now they want to discuss a different vector space where direction may not have a clear definition.

1. Matrices having a single row or column are referred to as vectors.

This is a bit more advanced than what you are probably studying. It basically comes down to how matrices actually arise. Once you fix a basis for your vector space, there is a bijective correspondence between linear transformations and matrices. Then all matrices arise as such. The proof involves taking a basis for the domain and then the columns (or rows) are the images under this map. Well, the image is an element of the codomain, i.e. an element of a vector space, so we can call it a vector.

This way, we can see that all columns of such a matrix is a vector of the codomain. For rows, now just switch the codomain and the domain.

1. I also watched a video in which at approximately 3:55 he says that a point in two dimensional real coordinate space is written in matrix form in LINEAR ALGEBRA.

This is going back to 1, where we are once again working in what appears to be $\mathbf{R}^2$. He says it is more common to write the vector $(5,0) \in \mathbf{R}^2$ as a column matrix instead of as a point notation. This is merely a naming or a left vs. right ($xA = b$ vs. $Ax = b$ if you will) and has nothing to do with whether it's a vector.

• Re 2: this has nothing to do with linear transformation or basis. The set of all (say real) matrices of fixes dimension $m \times n$ forms a vector space. What is said is that if $m=1$ (or also if $n=1$) then such a special matrix is called a vector. While a $13 \times 237$ matrix is usually in fact not called a vector, even though it is an element of a vector space. – quid Dec 25 '15 at 20:22
• For example, if you considered a $m \times 1$ column vector, then I claimed that you can realize it as a vector of $\mathbf{R}^m$ because it comes from a map from $\mathbf{R} \to \mathbf{R}^m$. – Future Dec 25 '15 at 20:24
• Yes you can. But this is not what is meant in the quoted source. – quid Dec 25 '15 at 20:29
• There was no quoted source for 2, so I merely justified how in general, one can say the columns or rows of a matrix are 'vectors' consistent with the definition. – Future Dec 25 '15 at 20:34
• My point is that there is no need at all for linear transformations to justify calling this a vector. Actually consistent with the start of your answer there is no need for any justification beyond that matrices of given dimensions form a vectorspace; thus according to you they are vectors anyway. – quid Dec 25 '15 at 20:54

A vector is an element of a vector space.

• Comments are not for extended discussion; this conversation has been moved to chat. – Daniel Fischer Dec 26 '15 at 18:30
• @DanielFischer, the comments were strictly about the correctness of the answer. This is not what comment removal by moderators is intended to accomplish. Please revert. – zyx Dec 26 '15 at 19:14

A vector is an element of a vector space.

That is not correct.

"Vector" is a term that was in use before the concept of vector space existed, is applied to things that are not elements of a vector space, and the elements of a given vector space are usually referred to by other names that depend on the space.

"Vector space" means a space with vector structure ( = operations of addition, subtraction, and scalar multiplication subject to the vector space axioms), not a space whose elements are necessarily called "vectors". Linear and vector-space structure are also used interchangeably for the same concept. This naming convention is the same as for topological spaces, metric spaces, Banach spaces, uniform spaces, and pretty much every other kind of structured space defined in theoretical mathematics. An X space is a space with X-structure, but is not necessarily a space whose elements are X's.

Although elements of a vector space can, in principle, be referred to as vectors, usually they have other preferred names, such as "points" or "functions" or "sections", that take precedence. Geometric arrows from one point to another are called vectors, but do not satisfy the vector space axioms due to lack of uniqueness of $0$. It is the displacements (equivalence classes) associated with geometric vectors that do form a genuine vector space. "Vector" also is used, both in mathematics and computer programming, to denote ordered $n$-tuples whose components all are the same type of object, and those vectors do not in general carry the algebraic operations needed to form a vector space.

• The title of the question is "What does vector mean in linear algebra?" (emphasis mine). – Rahul Dec 25 '15 at 19:47
• I think even in that context, elements of a lattice (no scalar multiplication) are called vectors. As far as I can see the linguistic arrow is in one direction only, from "element of vector space" to "vector" but not in general the reverse. Linear algebra also includes the use of what for purposes of this answer I called geometric vectors (drawn as arrows and called simply vectors) which are pairs of points in affine spaces rather than elements of genuine vector spaces with a distinguished origin. – zyx Dec 25 '15 at 19:49
• It is not even true that elements of a vector space are usually called vectors. Neither in abstract nor in many specific cases. – quid Dec 25 '15 at 20:14
• They can, in an act of logical-linguistic consistency, be called vectors, but usually are not. This was the purpose for the discussion of "other preferred names that take precedence" in paragraph 2. When you say this also is true in abstract, do you mean that elements of an abstract vector space are more often called "elements" than "vectors" (which I think is the case in linear algebra proofs, for example)? @quid – zyx Dec 25 '15 at 20:21
• Yes this is what I meant. Few people will say. Let $V$ be a vector space of dimension $n$ and let $v_1, \dots, v_{2n}$ be vectors in $V$ or something like that, except perhaps at a very basic level. – quid Dec 25 '15 at 20:27

The word Vector is used to name some closely related ideas - so close in fact that they can be regarded as having the same structure.

I was introduced to vectors as quantities which had magnitude and direction in two or three dimensions. Two dimensions for plane geometry in maths and three dimensions for forces and velocities and the like in physics. It seems from this description as if the magnitude and direction would be the mathematically important aspects. However, two different sets of ideas also come into play.

The first is that we can describe positions in space or on the plane using co-ordinates. So we can convert all our vectors and their properties into the language of co-ordinates. Once we have chosen a co-ordinate system we have a language for talking about vector quantities, and it turns out that vectors can be identified with ordered pairs or triples or quadruples etc depending on the dimension of the space in which we are working. So now, instead of magnitudes and directions, we have ordered $n$-tuples to describe the same thing.

The second insight is that we ought to be able to do geometry without having to choose a system of co-ordinates. Co-ordinates can be convenient in many cases, but they can also obscure the simplicity or pattern of what is happening. But to do anything serious with vectors we need to know their properties. It turns out that the fact that you can add vectors together, together with the possibility of multiplying them with scalars provides a rich structure which makes sense of the original vectors in two or three dimensions, and gives general results which apply in mathematically significant situations much more widely. Since the results we prove depend only on the basic structure, as in a great deal of mathematics we work more generally than our first example, and end up gaining a great deal from the generality. But we always have our first example there as a basic example.

It is this structural description which brings in the "linear algebra" title - because adding and multiplying by scalars are fundamental to linear systems in mathematics (and indeed in physics).

Hope that helps you to understand a bit why things seem more complicated just at the moment. I have got many answers to my questions. And after doing many questions in Linear algebra, I have compiled a few examples which I encountered. so these examples are representing how many types of "vectors" are used. please tell me if anyone of you wanna change anything...

UPDATE: 2nd May 2016

I just came across the paper I have uploaded here in my file. And I suddenly noticed the error. So I thought I should correct it here too. Number 2 and 4 are incorrect. Because if we V is a vector space over the field of real numbers and V is some subset of R or C then the subsets cannot be always called vector space over R :

1. If it does not contain additive inverse and zero (since V should be an Abelian group w.r.t. addition)

2. If it does not contain multiplicative inverse and 1. (since V should be closed under scalar multiplication)

So according to these two points (and other points of vector space), Q and N cannot be vector spaces and also their subsets. Z and C are vector spaces over R but if their subsets do not follow the above two points then they won’t be vector spaces.

• Number $2$ and number $4$ are incorrect. – Michael Albanese Jan 26 '16 at 7:53
• but why the subsets of real and complex numbers cannot be called vectors? – shivani Jan 26 '16 at 8:06
• Because they aren't. – Michael Albanese Jan 26 '16 at 9:18