I know how to prove by induction in general but in this task I don't even understand how to do and apply it, it's last task from an old exam:
$\mathbb{R}_{n}[x]= \left\{\sum_{k=0}^{n}p_{k}x^{k}: p_{k} \in \mathbb{R}\right\}$ is the vector space of all real polynomial $p$ of the degree $n_{p} \leq n$. And for every $j \in \left\{0,1,...,n\right\}$ there is a polynomial $P_{j} \in \mathbb{R}_{n}[x]$ given with degree $np_{j}=j$, that means $P_{j}$ has the shape $$P_{j}= \sum_{k=0}^{j}p_{jk}x^{k} \text{ with } p_{jk} \in \mathbb{R} \text{ for } k \in \left\{0,1,...,j\right\} \text{ and } p_{jj} \neq 0$$
Prove by induction that $\left\{P_{0},..., P_{n}\right\}$ is a basis of $\mathbb{R_{n}[x]}$, if $\mathbb{R}_{0}[x]$ is identified with $\mathbb{R}$.
If this was in my exam, I wouldn't know at all what to do and where to start. There are so so many different variables and attributes given... I would be very happy already if I knew how to do the start (I mean where you just show it for a specific $n$, the begin). Usually, induction proofs were easy when I had them in analysis classes but this is something very different and much more complicated I don't know how to start here? >.<
you need the set of $\{p_0,p_2,\cdots, p_n\}$ to span the space and to be linearly independent.
And since the space of $\mathbb R_n[x]$ has dimension $n+1$ and there are $n+1$ vectors in our proposed basis, you really only need to prove that the set is linearly independent.
Base case: $n=0$
$p_0 = c$ spans the set of real numbers
Suppose: $\{p_0,p_2,\cdots, p_n\}$ is a basis for $R_n[x]$
We must show that:
$\{p_0,p_2,\cdots, p_n, p_{n+1}\}$ is a basis for $\mathbb R_{n+1}[x]$
$p_{n+1}$ has an $x^{n+1}$ term that $\{p_0,p_2,\cdots, p_n\}$ do not.
$p_{n+1}$ cannot be formed by a linear combination of $\{p_0,p_2,\cdots, p_n\}$
$\{p_0,p_2,\cdots, p_n, p_{n+1}\}$ are a linearly indpendent set of vectors that spans $\mathbb R_{n+1}[x]$ and therefor form a basis for $\mathbb R_{n+1}[x]$
QED