# 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. – Ali Caglayan Oct 6 '14 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? – Student Oct 6 '14 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! – flawr Oct 6 '14 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. – voldemort Oct 6 '14 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. – voldemort Oct 6 '14 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. – Student Oct 6 '14 at 19:19