If $M, N$ are finite dimensional vector spaces with same dimension, then if $M$ is subset of $N$, then $M=N$. If $M, N$ are finite dimensional vector spaces with same dimension, then if $M$ is subset of $N$, then $M=N$.
I think i need to show that both vector spaces are spanned by the same bases in order to do this or to prove $N$ is subset of $M$? 
But i am not sure how to do this. 
Thanks

Let me reformulate the problem:

If $M$ is a (finite dimensional) subspace of a finite dimensional vector space $N$ and $\dim M=\dim N$, then $M=N$.

The clause that $M$ is finite dimensional is redundant, but it's not important, as you have it in your assignment.
Suppose $M\ne N$ and let $v\in N$, $v\notin M$. If $\{v_1,\dots,v_n\}$ is a basis for $M$, then $\{v_1,\dots,v_n,v\}$ is a linearly independent set in $N$: this is impossible, because a linearly independent sets in $N$ has at most $n=\dim N=\dim M$ elements.
