Let ( B_1 = left{(2,1,1,1),(1,1,1,1),(1,1,2,1)ight} hspace{2mm} )and ( hspace{2mm}B_2 =left{(2,1,2,2)ight}hspace{2mm} )be two subsets of( hspace{2mm} mathbb{R}^4, E_1 ) be a subspace spanned by( B_1, E_2 )be a subspace spanned by ( B_2 ), and L be a linear operator on ( mathbb{R}^4 ) defined by
( L(v)=(-w +4x-y+z,-w+3x,-w+2x+y,-w+2x+z)hspace{2mm},v=(w,x,y,z) )
(a) Show that ( B_1 ) is a basis for ( E_1 ) and ( B_2 ) is a basis for ( E_2 )
(b) Show that ( E_1 ) and ( E_2 ) are L-invariant. Find the matrices ( A_{1} =[L_{E_1}]_{B_1} ) and ( A_2=[L_{E_2}]_{B_2} )
