Question: > 
> *i)* What is the dimension of the linear space $\mathbb{R}^{n\times n}$
>  of all $n\times n$ matrices?
> 
> 
> *ii)* What is the dimension of the subspace of symmetric matrices in
>  $\mathbb{R}^{n\times n}$?
> 
> 
> 

**Def:** *The dimension of the linear space $V$ is defined as the maximum number of linearly independent vectors in $V$.*

I really don't understand att all how I can use that definition to answer *i)* and *ii)* above. How should I think through this?

Answer: 
You are trying to solve the differential equation 
$$\frac{d^2 y}{d x^2} + a \frac{d y}{d x} + b = 0,$$ which, on the level of power series is 

$$n (n-1) a\_n + a (n-1) a\_{n-1 }+ b a\_{n-2} = 0,$$ for $n \geq 2.$

Otherwise, if the roots of $P$ are real, this will be a linear combination of two exponentials, otherwise of sines and cosines.
Answer not using analysis.  

We look for formal series solutions of $P(D)y(x)=0$ in the form $y(x)=\sum\_{n=0}^\infty \frac{c\_n}{n!} x^n$ where $n!=1\cdot2\cdot...\cdot n$. Then $Dy(x)=\sum\_{n=0}^\infty \frac{c\_{n+1}}{n!} x^n$. The equation $P(D)y(x)=0$ is satisfied if and only if $c\_{n+2}+a\,c\_{n+1}+b\,c\_{n}=0$ for all nonegative integers $n$.  

The solution sequences of this ''recurrence'' equation are determined by their ''initial'' terms $c\_0,c\_1$. Hence they form a vector space of dimension 2.  

It is more convenient to express them in a different basis:
If $t^2+at+b=0$ has two roots $r,s$ in $k$, then use the sequences
$r^n, n\geq0$ and $s^n, n\geq0$. If $t^2+at+b=0$ has no roots in $k$, then it has two conjugate roots $r,\bar r$ in $\bar k$. A basis in $k[[x]]$ is then given by the sequences $r^n+\bar r^n, n\geq0$ and $r^{n+1}+\bar r^{n+1}, n\geq0$. Here I use one assumption in the form that $\bar k=k[q]$, where $q^2\in k$ and therefore elements $f$ of $\bar k$ satisfying $\bar f=f$ are in $k$. I also used the assumption that there are no double roots of $t^2+at+b=0$.  

Without wanting to use any analysis, let me indicate that $\sum\_{n=0}^\infty \frac{r^n}{n!}x^n$ is the power series of an exponential function.