Vectors, vector spaces, and linear algebra

Question

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?

Answer 1

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 $(0,0)$.

Answer 2

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 $u$.