The representation of vectors by any basis is unique, as established in the discussion. The proof involves expanding a vector \( \vec{w} \) into a basis set \( \{\hat{w}_i\} \) using two different sets of coefficients, \( a_i \) and \( b_i \). By demonstrating that \( \sum_i (a_i - b_i)\hat{w}_i = 0 \) and leveraging the independence of the vectors, it is concluded that \( a_i = b_i \) for all \( i \). This confirms the uniqueness of representation in linear algebra.
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