Basis of dual space. Let $V$ be a finite-dimentional vector space over the field $F$, and let $B=\{ \alpha_1 ,\alpha_2,....,\alpha_n\}$ be a basis for $V$. Then there is a unique
dual basis $B^*=\{ f_1,f_2,....,f_n\} $ for $V^*$ such that $f_i(\alpha_j)=\delta_{ij}.$

My question is why there is a unique basis which is dual to $B$.(why unique?)
I know that $f_i(\alpha_j)=\delta_{ij}$ is a basis for dual space but there can be other basis for the dual space!

Thanks!
Well, if you already found one then observe each $\;f_i\in V^*\;$ is uniquely determined by its values on the basis $\;B\;$ , and this unique determination corresponds with the basis elements indexwise.

Thus, any other such basis must contain linear functionals $\;g_i\in\ V^*\;$ each of which agree with a correspondent $\;f_i\;$ on the basis $\;B\;$ indexwise and thus it is the same dual basis as then $\;g_i=f_i\;$ for all $\;i\;$ .
Because a dual basis is the basis of dual space, it spans the dual space.
We can suppose $B=\{v_1,...,v_n\}$ is a basis of the dual space, and because it is the basis of the dual space, $B$ spans $V'$.
And the dual space $V'$ denotes the set of all linear functionals that map some elements in the space $V$ to a scalar.
Therefore, $B$ can represent any linear functional in $V'$. For an arbitrary linear functional $T$ in the dual space $V'$, $T$ can be represented by the basis of dual spaces and it must be unique because every different linear combination of linearly independent list is unique, and without loss of generality we can also say that any linear functionals represented by a linear combination of the basis (that are linearly independent) of $V'$ are also unique.