How determine if a set form a basis for a vector space In the book I am studying, the definition of a basis is as follows:  
If $V$ is any vector space and $S= \{ \textbf{v}_1,...,\textbf{v}_n \}$  is a finite set of vectors in $V$, then $S$ is called a basis for $V$ if the following two conditions hold:
(a)  $S$ is lineary independent
(b) $S$ spans $V$  
I am currently taking my first course in linear algebra and something about the notion of a basis bugs me. The reason is that I feel like one would need a basis to investigate if the conditions (a) and (b) holds, but how do one find this first basis?   
As a dummy example from the book, the author shows that the standard basis for $R^3$ is in fact a basis for $R^3$ in (roughly) the folowing way:  
"We need to show that
$\textbf{i} = (1,0,0),\quad \textbf{j} = (0,1,0), \quad \textbf{k} =  (0,0,1)$
are lineary independent and that they span $R^3$.  
To show that they are lineary independent, we note that if they are lineary dependent, then the equation
$t_1(1,0,0) + t_2(0,1,0) + t_3(0,0,1) = (0,0,0)$
would be satisfied by some $t_r \neq 0$. But equating corresponding components shows that this is equivalent to saying that
$t_r = 0, \quad t_r\neq 0$
which is a contradiction.  
To show that condition (b) holds, we note that an arbitrary vector $\textbf{v}$ in $R^3$ can be written as
v $=t_1(1,0,0) + t_2(0,1,0) + t_3(0,0,1)$"  
Ok, so the thing that confuses me is that in this example, to show that the standard basis form a basis, we have used... the standard basis? I mean, when stating that $\textbf{i} = (1,0,0)$ we are basically saying that $\textbf{i}$ actually $\textbf{is}$ one of the unit vectors which forms the standard basis right? But then we are showing that the standard basis is a basis by assuming that it is a basis from the beginning, which does not make sense.
I am going to try to summarise this rather vague post with some concrete questions, but feel free to adress something that you feel is relevant to my post.  
1.) Is it possible to prove that a standard basis in fact form a basis, or should one take this as kind of an axiom?  
2.) If (1.) is true, what does the components of a vector actually mean before you have proven that there exist a "first" basis?
 A: It is possible to prove that standard basis forms a basis. First of all standard basis is the most obvious way of expressing that vectors are linearly independent. $\textbf{i} = (1,0,0),\; \textbf{j} = (0,1,0), \; \textbf{k} =  (0,0,1)$ are linearly independent because $t_1(1,0,0) + t_2(0,1,0) + t_3(0,0,1) = (0,0,0)$ iff $t_1=t_2=t_3=0$. (You can think them as columns of identity matrix to see that it is the only possible case)
Additionally,
$$t_1\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}+t_2\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}+t_3\begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}=\begin{pmatrix} t_1 \\ t_2 \\ t_3 \end{pmatrix}$$
So linear combinations of these vectors are determined by the values of $t_1,t_2,t_3$ and the arbitrary values of $t_1,t_2,t_3$ forms $\mathbb R^3$.
A: *

*usually, vector spaces are not given by a base and not all of them has a base referred to as the standard base.

*Any set which you can prove is independent and it's span is the whole space, is a basis.  This also holds for the standard basis. You should verify to yourself that you know how to prove that $ e_1 ,\dots,e_n $  is a base of  $\mathbb R ^n$.

*Usually, both criteria are proven by solving a set of linear equations or a least proving that the set of equations have a unique solution for the representation of each vector in the space.  

*There is no special meaning for the component on a certain coordinate of a vector. The usual geometric meaning appears on 2-d and 3-d spaces of real valued numbers in relation to their standard base. 
A: In fact, it can be proved that every vector space has a basis by using the maximal principle; you may check, say Friedberg's linear algebra book. To find out a concrete basis for a vector space, we need the characterizing conditions.
The coordinate vector of a vector is defined in terms of a chosen basis, so there is no such thing as talking of coordinates of a vector without referring to a basis.
A: You can think of it as a set $\{\mathbf{i},\mathbf{j},\mathbf{k}\}$ that you want to prove certain properties of. After you've proven these properties you know that you can call it a basis. Since it is the first set you would intuitively choose to check the properties for, you choose to call it the standard basis. The coordinates you can see as being defined basis-free as an algebraic object acting in certain ways under operations. After you then have a standard basis, you then conclude that there is a natural way of associating the basis vectors with the coordinates.
