Let B={b1,b2,b3}\mathcal{B} = \{\mathbf{b}_1, \mathbf{b}_2, \mathbf{b}_3\}B={b1,b2,b3} be a basis for R3\mathbb{R}^3R3. If w=c1b1+c2b2+c3b3=d1b1+d2b2+d3b3\mathbf{w} = c_1\mathbf{b}_1 + c_2\mathbf{b}_2 + c_3\mathbf{b}_3 = d_1\mathbf{b}_1 + d_2\mathbf{b}_2 + d_3\mathbf{b}_3w=c1b1+c2b2+c3b3=d1b1+d2b2+d3b3, then:
c1=d1,c2=d2,c3=d3c_1 = d_1, c_2 = d_2, c_3 = d_3c1=d1,c2=d2,c3=d3
The coefficients are never unique
(c1,c2,c3)(c_1, c_2, c_3)(c1,c2,c3) and (d1,d2,d3)(d_1, d_2, d_3)(d1,d2,d3) may differ
c1+d1=c2+d2=c3+d3c_1 + d_1 = c_2 + d_2 = c_3 + d_3c1+d1=c2+d2=c3+d3