Am I understanding vectors and matrices properly? So, here is my understanding of a Vector:  
A vector is an ordered set of real numbers that lie in the space $R^n$ where $n$ is the size of the vector. 
So if $n$ equals 4, the vector is of size 4.  
I understand matrices to be a set of vectors - row vectors and column vectors.
So, given a matrix $A_3\times_2$, the row vectors are in the space $R^3$ because each row has only 3 components and the columns are in the space $R^2$ because each column can only have 2 components.  
And finally here is my understanding of of $R^n$:
It means a an ordered set of $n$ numbers such that each one of the numbers is a real numbers. Each ordered set is called a $tuple$ and a tuple can not have more than $n$ elements/components in it. 
In the most basic way, is my understanding correct?
I was very bad at math in high school and now I am taking an effort to make it up and improve my math skills.
 A: *

*The word "size" is not really used for the length of a vector, (normally we would just go all the way and say "number of components" or "coordinates" of the vector). We often say that the dimension of $R^n$ is $n$. 

*Note that, in higher generality, a vector space does not have to be presented in the form $R^n$ (i.e. it does not have to be comprised of coordinate vectors), and it does not have to be a space over the real numbers. (It could be a space defined over complex numbers, rational numbers, finite fields, etc. as the set of scalars.) A space simply satisfies some list of abstract axioms.

*The size, also called magnitude, of a vector is measured by some kind of norm $\|\cdot\|$, which is often induced by some sort of inner product $\langle\cdot,\cdot\rangle$. For instance, in the Cartesian plane ${\bf R}^2$ with the standard inner product, the size of the vector $(3,4)$ is $\sqrt{3^2+4^2}=5$.

*Indeed an $m\times n$ matrix has $n$ column vectors and $m$ row vectors. Sometimes a matrix is thought of as an array and not an ordered set of vectors. 

*The key use of matrices is that we can multiply them together, and every linear map on a vector space can be encoded as multiplication by some matrix (using a basis to represent the vectors as coordinate vectors); this will probably turn up later in your study.

*The notation ${\bf R}^n$ does not itself denote an ordered $n$-tuple. Rather it denotes the space of all ordered $n$-tuples with real number entries, where addition is defined "componentwise" and scalar multiplication is defined in the obvious way, etc.

*More generally, vector spaces do not need to be finite-dimensional, but infinite-dimensional vector spaces verge into different territories of mathematics (functional analysis, set theory, ..)
A: You are absolutely right that $\mathbb{R}^n$ makes up an $n$-dimensional vector space, whose vectors are lists of numbers of length $n$: $(x_1,x_2,\ldots,x_n) \in \mathbb{R}^n$. Indeed, matrices are also vectors. Say you have a 3-by-3 matrix, you could write down the first row, then the second and then the third, all in a row. This would give a list of nine numbers. So the 3-by-3 matrices can be thought of as vectors in $\mathbb{R}^9$.
However, vectors are very general objects. A vector is something that belongs to what we call a vector space. Any collection of objects that pass some tests, comply with some axioms, is called a vector space, and its elements are called vectors. Follow this link to find the tests that the things need to pass.
The abstract definition of a vector space was, I think, influenced by the idea of vectors that we meet in school and in physics, i.e. things with magnitude and direction, e.g. velocity, acceleration or force vectors. They can be added, subtracted, and scaled up and down. But there are lots of objects with these properties.
The set of complex numbers $x + iy$, where $x$ and $y$ are real numbers is a two dimensional $\mathbb{R}$-vector space. The set of numbers $a+b\sqrt{2}$, where $a$ and $b$ are fractions, is a two dimensional $\mathbb{Q}$-vector space. There are more exotic example: the set of real valued functions on the closed interval $0 \le x \le 1$ is an infinite dimensional vector space. Given $\operatorname{f},\operatorname{g} : [0,1] \to \mathbb{R}$, we can define $\operatorname{f} + \operatorname{g}$ by $(\operatorname{f}+\operatorname{g})(x) = \operatorname{f}(x)+\operatorname{g}(x)$ for all $0 \le x \le 1$, the product $(\operatorname{fg})(x) = \operatorname{f}(x)\cdot\operatorname{g}(x)$ and scaler multiplication by $(\lambda\operatorname{f})(x) = \lambda \cdot \operatorname{f}(x).$
