Isomorphism between vector space and dual space

Question

My teacher told me if two finite dimensional vector space have same dimension, then they're isomorphic.

According to this, shouldn't every FDVS be isomorphic to its dual?

Answer 1

A finite dimensional vector space $V$ is always isomorphic to its dual space $V^*$. This is a consequence of the fact that $V$ and $V^*$ have the same dimension.

We can easily see this finding a bases for $V^*$. If $(\vec e_i)$ is the canonical basis of $V$ such that any vector can be expressed as $\vec v=v_1 \vec e_1+v_2 \vec e_2 +\cdots +v_n\vec e_n$.

Consider the linear functionals $f_i$ defined by $ f_i (\vec v)= v_i$.

So, for the vectors of the basis we have $f_i(\vec e_j)=\delta_{i,j}$ and any linear functional can be expressed as a linear combination of $(f_i)$, so $(f_i)$ is a basis of $V^*$.