Question 4.7.3: Finding a Coordinate Matrix Relative to a Nonstandard Basis ...
Finding a Coordinate Matrix
Relative to a Nonstandard Basis
Find the coordinate matrix of x = (1, 2, -1) in R³ relative to the (nonstandard) basis
B´ = \left\{u_1, u_2, u_3\right\} = {(1, 0, 1), (0, -1, 2), (2, 3, -5)}.
Begin by writing x as a linear combination of u_1, u_2, and u_3.
x = c_1u_1 + c_2u_2 + c_3u_3
(1, 2, -1) = c_1(1, 0, 1) + c_2(0, -1, 2) + c_3(2, 3, -5)
Equating corresponding components produces the following system of linear equations and corresponding matrix equation.
\left [ \begin{matrix} 1 & 0 & 2 \\ 0 & -1 & 3 \\ 1 & 2 & -5 \end{matrix} \right ] \left [ \begin{matrix} c_1 \\ c_2 \\ c_3 \end{matrix} \right ] = \left [ \begin{matrix} 1\\ 2 \\3 \end{matrix} \right ]
The solution of this system is c_1 = 5, c_2 = -8, and c_3 = -2. So,
x = 5(1, 0, 1) + (-8)(0, -1, 2) + (-2)(2, 3, -5)
and the coordinate matrix of x relative to B´ is