A First Course in Linear Algebra · Problem RQ-VR-2

The core concept here is change of basis: you're finding the coordinates of a vector with respect to a non-standard basis $B$. Rather than the usual standard basis, you need to express the given vector as a linear combination of $B$'s vectors—those coefficients are exactly what $\rho_B$ (the coordinate map) returns. Set up a system where $c_1 \mathbf{b}_1 + c_2 \mathbf{b}_2 + c_3 \mathbf{b}_3$ equals your target vector, then solve for $c_1, c_2, c_3$.

The core concept here is change of basis: you're finding the coordinates of a vector with respect to a non-standard basis $B$. Rather than the usual standard basis, you need to express the given vector as a linear combination of $B$'s vectors—those coefficients are exactly what $\rho_B$ (the coordinate map) returns. Set up a system where $c_1 \mathbf{b}_1 + c_2 \mathbf{b}_2 + c_3 \mathbf{b}_3$ equals your target vector, then solve for $c_1, c_2, c_3$.