Are the coefficients of a vector according to a basis unique? If I have a vector space $V$ ( of dimension $n$ ) over real numbers such that $\{v_1,v_2...v_n\}$ is the basis for the space ( not orthogonal ). Then I can write any vector $l$ in this space as $l=\sum_i\alpha_iv_i$. Here $\alpha_1,\alpha_2...\alpha_n$ are the coefficients that define the vector  $l$ according to this basis. Can another set of coefficients $\beta_1,\beta_2...\beta_n$ give the same vector $l$ ? If the basis was orthogonal the answer would be no, but I can't prove for a non orthogonal basis.

They cannot.  Suppose so.  Then $\sum \alpha_i v_i = \sum \beta_i v_i$.  This implies:
$$0 = \sum (\alpha_i - \beta_i) v_i$$
which violates the basis being a set of linearly independent vectors.
