Understanding Vector Spaces I'm struggling to grasp the concept of a vector space. In my notes, some examples of vector spaces are:
$1$) $V =$ set of all $n \times n$ matrices, $F = \mathbb{R}$
$2$) $V =$ set of all real valued functions on $[0, 1], F = \mathbb{R}$
$3$) $V =$ set of all continuous functions on $(0, 1)$
Do vector spaces only contain vectors? If they do, how can $1$) be true since it contains all square matrices (not vectors).
 A: You should view the term "vector" abstractly. 
They can be arbitrarily objects as long as they satisfy the axioms. 
For example, for your first example, for matrices over real number:
It is associative:
$$(A+B)+C=A+(B+C)$$
It is commutative:
$$A+B=B+A$$
It has identity element. The zero matrix. $A+0=A$, here the $0$ is the matrix where all elements are $0$.
It has an inverse. $A-A=0$.
You can check the other axioms that has to be safisfied here and verify that those properties are satisfied.
A: Matrices are vectors!
We say that $e_1 = \begin{bmatrix} 1&0\\0&0\end{bmatrix}, e_2 = \begin{bmatrix} 0&1\\0&0\end{bmatrix},e_2 = \begin{bmatrix} 0&0\\1&0\end{bmatrix},e_4 = \begin{bmatrix} 0&0\\0&1\end{bmatrix}$ is the basis of our vector space.  And any $2\times 2$ matrix is a linear combination of these elements.
The elements in a vector space are "vectors", even if they don't conform to the geometric properties of vectors in $\mathbb R^2, \mathbb R^3$
Any collection of objects with an addition property and a scalar multiplication property and conform to the various axioms (closure under addition, closure under scalar multiplication, zero vector, additive inverse, distribution of scalar multiplication over addition, etc.) is a vector space.
Real numbers are vectors, too.  They meet all of there requirements of the definitions.
If we consider the set of functions with domain $\{1,2,3\}$
then $(f(1), f(2), f(3))$ behaves just like a vector in $\mathbb R^3$
And, we can abstract to the set of real valued functions defined everywhere over the interval [0,1], and $f(t)+g(t)$ is a real valued function, and $af(t)$ is a real valued function, etc.  
When you are ready to get your mind bent, differentiation is a linear transformation of a vector space functions.
A: The beauty of mathematics is abstraction, but we often pay a price for this abstraction. For a concrete example, let's talk about what "$3$" means. When first teaching someone to count and add, we focus on three stones, three fingers, three dots on a page, three books, etc. All of these are fundamentally different objects, but we can abstract all of them with the notion of "$3$." Since we are now talking about a supremely abstract concept of "$3$", we can make statements like
$$3 + 4 =7$$
without needing to separately show that this is true for


*

*three stones and four stones make seven stones

*three dots and four dots make seven dots

*three fingers and four fingers make seven fingers


and repeating this ad nauseum for every possible group of things that someone might possibly want to count. Instead, we just show that $3+4=7$ and rest assured that this applies equally well to every possible collection of objects that someone might want to apply it to. 
Similarly with linear algebra, we have many different things that we can view as vectors: a physicist might say that a vector is an arrow (perhaps representing magnitude and direction of a force) while a computer scientist might say that a vector is a list of numbers that should be considered a single object. Rather than treat everything like this as a completely separate field of object (up arrow plus down-right arrow gives ..., or [\$100, 3 people] + [-\$42, 16 people] = ...) we instead analyze all of them.  To a mathematician, a vector/vector space is...anything that behaves like vectors with reasonable notions of addition and scalar multiplication, just like how "$3$" is three fingers, three dots, three stones, ...
This is why the definition of a vector / vector space seems so abstract: we want to be able to make statements about anything that behaves like a vector. This is just like the jump from thinking about three fingers and four fingers making seven fingers to thinking "$3+4=7$". Instead of analyzing what happens when we combine arrows or combine lists, we analyze what happens to any vector space under any linear transformation. This way we can prove general results about all vector spaces, without needing to have a separate field of study for every single possible "vector-ish" thing. 
The youTube channel 3 Blue 1 Brown has a nice video that graphically illustrates this concept 
A: As long as it obeys the axioms, it can form a vector space. Another example is the solution space of a differential equation. If you have solutions $y_1$ and $y_2$, their linear combination $\alpha_1y_1+\alpha_2y_2$ for any scalar constants $\alpha_1$ and $\alpha_2$ is also a valid solution of the differential equation. Maxwell's equations of Electromagnetism in physics are a prime example where this holds.
A: A vector space V is a set that is closed under finite vector addition and scalar multiplication. There are 8 axioms to be respected in order to be a vector space (commutativity, distributivity, for example).
A matrice is made out of vectors. You can write a vector in a matrice as a row or a column. For example the matrice 1x2 (1 row 2, columns) \begin{bmatrix} 1&0\\\end{bmatrix} 
is made of the vector (1,0). 
A: Items that we have :

V is a set
$+$ is an internal law on the elements of V
  $\mathbb{F}$ is a field
$.$ is an external application from $\mathbb{F} \times V \to V$ 

By definition, (V, $+$, $.$) is an $\mathbb{F}$-vector space if all these properties are validated :  

(a) (V, +) is a commutative group;  
(b) for all $\lambda_1 \in \mathbb{F}, \lambda_2 \in \mathbb{F}, v_1 \in V, v_2 \in V$:
  $(\lambda_1 + \lambda_2).v_1 = \lambda_1.v_1 + \lambda_2.v_1$
  $\lambda_1.(v_1+v_2) = \lambda_1.v_1 + \lambda_1.v_2$    
(c) for all $v \in V$, $1_\mathbb{F}.v=v$  
(d) for all $\lambda \in \mathbb{F}, \mu \in \mathbb{F}, v \in V$:
  $\lambda.(\mu.v)=(\lambda\times\mu).v $  

From your examples, let's focus on the matrices :

(1) Is the set of $n\times n$ matrices is a $\mathbb{R}$-vector space?
(a) is ok : the sum of two $n\times n$ matrices is an $n\times n$ matrix, the addition is commutative and associative, the matrix with zeros everywhere is in the set and is a neutral element for $+$; every $n\times n$ matrix has an opposite $n\times n$ matrix in the set (their sum is equal to the $n\times n$ zero matrix).  
To check (b), (c) and (d), we must be sure that we understand how the "multiplication by a scalar" operation works.
  If I write $A=(a_{i,j})$, then $\lambda.A=(\lambda\times a_{i,j})$
  Both $A$ and $\lambda.A$ are $n\times n$ matrices.
(b), (c), (d) are ok (check it).  

With that, the set of $n\times n$ matrices is a $\mathbb{R}-$vector space : you can add/substract matrices, you can multiply a matrix by a number, and this goes smoothly.  
Please note that multiplying a matrix by another matrix has not been described at all in this process. This is a completely different topic.

In the same way :
  (2) The set of functions $\{f:[0,1]\to\mathbb{R}\}$ is an $\mathbb{R}-$vector space : you can add/substract functions, you can multiply a function by a number, and this goes smoothly.
(3) The set of functions $\{f:[0,1]\to\mathbb{R} | f$ is continuous on $[0,1]\}$ is an $\mathbb{R}-$vector space : you can add/substract continuous functions and still have a continuous function, you can multiply a function by a number and still have a continuous function, and this goes smoothly.  

Hope this helps!
