Question

Let be a basis of\({\mathbb{R}^n}\). .Produce a description of an \(B = \left\{ {{{\bf{b}}_{\bf{1}}},....,{{\bf{b}}_n}\,} \right\}\)matrix A that implements the coordinate mapping \({\bf{x}} \mapsto {\left( {\bf{x}} \right)_B}\). Find it. (Hint: Multiplication by A should transform a vector x into its coordinate vector \({\left( {\bf{x}} \right)_B}\)). (See Exercise 21.)

Short answer

\(A = P_B^{ - 1}\)

Step-by-step solution

Find the value of x

The coordinate of x relative to B can be expressed as follows:

\(\begin{array}{c}{\bf{x}} = {c_1}{{\bf{b}}_1} + {c_2}{{\bf{b}}_2} + ... + {c_3}{{\bf{b}}_n}\\ = {P_B}{\left( {\bf{x}} \right)_B}\end{array}\)

Find matrix A

The B coordinate vector is given as:

\({\left( {\bf{x}} \right)_B} = P_B^{ - 1}{\bf{x}}\)

So, the desired matrix \(A = P_B^{ - 1}\).