WeBWorK Standalone Renderer

(a) Find the coordinate vector of $x = \displaystyle\left[\begin{array}{c} 3 \cr 5 \cr 0 \cr \end{array}\right]$ with respect to the ordered basis

$E = \left\lbrace \displaystyle\left[\begin{array}{c} 1 \cr 3 \cr 6 \cr \end{array}\right] , \displaystyle\left[\begin{array}{c} 0 \cr 1 \cr -7 \cr \end{array}\right] , \displaystyle\left[\begin{array}{c} 0 \cr 0 \cr 1 \cr \end{array}\right] \right \rbrace$

of ${\mathbb R}^3$ :

$[x]_{E} =$

$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$		$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$

(b) Let $F_1$ be the ordered basis of ${\mathbb R}^2$ given by

$F_1 = \left\lbrace \displaystyle\left[\begin{array}{c} 3 \cr -1 \cr \end{array}\right], \displaystyle\left[\begin{array}{c} 1 \cr 5 \cr \end{array}\right] \right \rbrace$

and let $F_2$ be the ordered basis given by

$F_2 = \left\lbrace\displaystyle\left[\begin{array}{c} -2 \cr 3 \cr \end{array}\right], \displaystyle\left[\begin{array}{c} 1 \cr -2 \cr \end{array}\right] \right\rbrace$
Find the transition matrix $P_{F_2 \leftarrow F_1}$ such that $[x]_{F_2} = P_{F_2 \leftarrow F_1} [x]_{F_1}$ for all $x$ in ${\mathbb R}^2$ :

$P=$

$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$			$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$

You can earn partial credit on this problem.