(related to Lemma: Uniqueness Lemma of a Finite Basis)
Let \(v\) be a vector in a vector space \(V\) with the basis \(B\).
If \(B\) is finite, then we have \(B=\{b_1,\ldots, b_n\}\). Suppose that there are two different representations \[\begin{array}{ccl} v&=&\alpha_1b_1+\ldots+\alpha_nb_n,\\ v&=&\beta_1b_1+\ldots+\beta_nb_n.\\ \end{array}~~~~~~~~~~~~~~~~~( * )\] Then we have \[\begin{array}{ccl} 0&=&(\alpha_1-\beta_1)b_1+\ldots+(\alpha_n-\beta_n)b_n, \end{array}\] Because the representations \( ( * ) \) are different by assumption, there is at least one \(i\) for which \(\alpha_i\neq \beta_i\). But this would mean that \(\{b_1,\ldots, b_n\}\) are linearly dependent, in contradiction to the definition of a basis.
