( newcommand{vecs}[1]{overset { scriptstyle rightharpoonup} {mathbf{#1}} } )
( newcommand{vecd}[1]{overset{-!-!rightharpoonup}{vphantom{a}smash {#1}}} )
( newcommand{id}{mathrm{id}}) ( newcommand{Span}{mathrm{span}})
( newcommand{kernel}{mathrm{null},}) ( newcommand{range}{mathrm{range},})
( newcommand{RealPart}{mathrm{Re}}) ( newcommand{ImaginaryPart}{mathrm{Im}})
( newcommand{Argument}{mathrm{Arg}}) ( newcommand{norm}[1]{| #1 |})
( newcommand{inner}[2]{langle #1, #2 rangle})
( newcommand{Span}{mathrm{span}})
( newcommand{id}{mathrm{id}})
( newcommand{Span}{mathrm{span}})
( newcommand{kernel}{mathrm{null},})
( newcommand{range}{mathrm{range},})
( newcommand{RealPart}{mathrm{Re}})
( newcommand{ImaginaryPart}{mathrm{Im}})
( newcommand{Argument}{mathrm{Arg}})
( newcommand{norm}[1]{| #1 |})
( newcommand{inner}[2]{langle #1, #2 rangle})
( newcommand{Span}{mathrm{span}}) ( newcommand{AA}{unicode[.8,0]{x212B}})
( newcommand{vectorA}[1]{vec{#1}} % arrow)
( newcommand{vectorAt}[1]{vec{text{#1}}} % arrow)
( newcommand{vectorB}[1]{overset { scriptstyle rightharpoonup} {mathbf{#1}} } )
( newcommand{vectorC}[1]{textbf{#1}} )
( newcommand{vectorD}[1]{overrightarrow{#1}} )
( newcommand{vectorDt}[1]{overrightarrow{text{#1}}} )
( newcommand{vectE}[1]{overset{-!-!rightharpoonup}{vphantom{a}smash{mathbf {#1}}}} )
( newcommand{vecs}[1]{overset { scriptstyle rightharpoonup} {mathbf{#1}} } )
( newcommand{vecd}[1]{overset{-!-!rightharpoonup}{vphantom{a}smash {#1}}} )
(newcommand{avec}{mathbf a}) (newcommand{bvec}{mathbf b}) (newcommand{cvec}{mathbf c}) (newcommand{dvec}{mathbf d}) (newcommand{dtil}{widetilde{mathbf d}}) (newcommand{evec}{mathbf e}) (newcommand{fvec}{mathbf f}) (newcommand{nvec}{mathbf n}) (newcommand{pvec}{mathbf p}) (newcommand{qvec}{mathbf q}) (newcommand{svec}{mathbf s}) (newcommand{tvec}{mathbf t}) (newcommand{uvec}{mathbf u}) (newcommand{vvec}{mathbf v}) (newcommand{wvec}{mathbf w}) (newcommand{xvec}{mathbf x}) (newcommand{yvec}{mathbf y}) (newcommand{zvec}{mathbf z}) (newcommand{rvec}{mathbf r}) (newcommand{mvec}{mathbf m}) (newcommand{zerovec}{mathbf 0}) (newcommand{onevec}{mathbf 1}) (newcommand{real}{mathbb R}) (newcommand{twovec}[2]{left[begin{array}{r}#1 #2 end{array}right]}) (newcommand{ctwovec}[2]{left[begin{array}{c}#1 #2 end{array}right]}) (newcommand{threevec}[3]{left[begin{array}{r}#1 #2 #3 end{array}right]}) (newcommand{cthreevec}[3]{left[begin{array}{c}#1 #2 #3 end{array}right]}) (newcommand{fourvec}[4]{left[begin{array}{r}#1 #2 #3 #4 end{array}right]}) (newcommand{cfourvec}[4]{left[begin{array}{c}#1 #2 #3 #4 end{array}right]}) (newcommand{fivevec}[5]{left[begin{array}{r}#1 #2 #3 #4 #5 end{array}right]}) (newcommand{cfivevec}[5]{left[begin{array}{c}#1 #2 #3 #4 #5 end{array}right]}) (newcommand{mattwo}[4]{left[begin{array}{rr}#1 amp #2 #3 amp #4 end{array}right]}) (newcommand{laspan}[1]{text{Span}{#1}}) (newcommand{bcal}{cal B}) (newcommand{ccal}{cal C}) (newcommand{scal}{cal S}) (newcommand{wcal}{cal W}) (newcommand{ecal}{cal E}) (newcommand{coords}[2]{left{#1right}_{#2}}) (newcommand{gray}[1]{color{gray}{#1}}) (newcommand{lgray}[1]{color{lightgray}{#1}}) (newcommand{rank}{operatorname{rank}}) (newcommand{row}{text{Row}}) (newcommand{col}{text{Col}}) (renewcommand{row}{text{Row}}) (newcommand{nul}{text{Nul}}) (newcommand{var}{text{Var}}) (newcommand{corr}{text{corr}}) (newcommand{len}[1]{left|#1right|}) (newcommand{bbar}{overline{bvec}}) (newcommand{bhat}{widehat{bvec}}) (newcommand{bperp}{bvec^perp}) (newcommand{xhat}{widehat{xvec}}) (newcommand{vhat}{widehat{vvec}}) (newcommand{uhat}{widehat{uvec}}) (newcommand{what}{widehat{wvec}}) (newcommand{Sighat}{widehat{Sigma}}) (newcommand{lt}{<}) (newcommand{gt}{>}) (newcommand{amp}{&}) (definecolor{fillinmathshade}{gray}{0.9})