# Vectors, vector spaces, and linear algebra

I have a few doubts regarding vectors and linear algebra in general:

1. What is the formal definition of a vector?
2. When we say $\Bbb R^n$, do we mean the set of all column matrices with $n$ entries,or all row matrices with n entries, or all ordered $n$ tuples that is $(a_1,.....,a_n)$ ?
3. If all members of a vector space are vectors, since $\Bbb R$ is a vector space, then it is implied that all real numbers are vectors. Is my understanding correct ?
4. Geometrically, can we treat vectors as free or fixed? What is the correct convention?
• Real numbers are single component vectors. Oct 6, 2014 at 19:37

1. A vector is an element of a vector space

2. We usually mean ordered $n$ tuples .Though, we sometimes want the other descriptions- depending upon the context. All these spaces are naturally isomorphic (as vector spaces).

3. Real numbers are definitely vectors- as they are members of a $1$ dimensional vector space.

4. We usually treat vectors as "fixed". For example, the vector $(1,0)$ in $\mathbb{R^2}$ can be pictured as a vector whose "tip" is the point $(1,0)$ and "tail" is the point $(0,0)$.

• voldemort, when you say "usually treat", what do you mean? Is the convention flexible? Also, I have an intuitive grasp of what isomorphic means, but as I am a first year undergrad, could you please give me a rigorous definition? Oct 6, 2014 at 19:11
• It does not really matter if you write down the vector as a row or column, this is just a notation which do vary a lot depending on the teacher, but the object itself is the same, the notation is just a way of representing the object. You could even write down a vector in different polar forms but the object is still the same. I'd like to add that I do not like the terms of free and fixed since a vector is everywhere but the arrows we all like to draw as visualization is just another representation but not the object itself! Oct 6, 2014 at 19:13
• Usually treat means that in most contexts this is the meaning you would find. In math sometimes some people use a different convention- and to be safe I use the term "usually treat". Isomorphic means they have the same structure as vector spaces- i.e. there exists a one-one, onto linear map which is invertible between the two spaces. Oct 6, 2014 at 19:13
• @flawr: Exactly- the spaces are isomorphic. Most people that I know prefer to think $\mathbb{R^n}$ as $n$ tuples, but the other two approaches are the same anyway. Oct 6, 2014 at 19:14
• @flawr: Hmmmm aren't the 2 objects different? For instance, you cannot say that a 2-by-1 matrix is the same as a 1-by-2 matrix, can you? Voldemort's way of explaining it seems to make sense, as he acknowledges the isomorphism between the two spaces, but considers them as different objects. Oct 6, 2014 at 19:19
1. We say "vector" instead of "element of a vector space", especially by tradition in lessons on abstract vector spaces. We could also say "point", or "chair" if we wished.
2. When we say $$\mathbb{R}^n$$, we mean $$n$$-tuples.
3. Your understanding is correct, if you agree with my point n°1.
4. Geometrically, let us place ourselves in $$\mathbb{R}^2$$, "the plane": on your drawing, you can interpret the elements of $$\mathbb{R}^2$$ as you wish, as long as it makes sense. For me, I can represent $$(2,3)$$ by a point $$A$$, "point $$A$$ with coordinates $$2$$ and $$3$$". If I am interested in the line $$D$$ of equation $$y=5-x$$, for example because it is the tangent to the function $$f:\mathbb{R} \to \mathbb{R}, x\mapsto 3-\sin(x-2)$$ at A, I will more naturally talk about $$(1,-1)$$ as the vector $$u=(1,-1)$$, which is a directing vector of $$D$$. About $$D$$ I will say that it is the affine line of $$\mathbb{R}^2$$, $$D=A+\mathbb{R}.u$$, and I will obviously conceive of it as a set of points. And i will conceive $$\mathbb{R}u$$ as the vector line passing through $$u$$.