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}...

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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} )

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Solution a Show that B1 is a basis for E1 and B2 is a basis for ullet hspace2mmForhspace2mmE2 implies E1spanlefteginpmatrix2 1 1 1endpmatrixeginpmatrix1 1 1 1endpmatrixeginpmatrix1 1 2 1endpmatrix ight Check eginpmatrix2111 1111 1121endpmatrixsim eginpmatrix1000 1111 0010endpmatrix implies LI Therefore B is a basis for E1 and B2 is basis for E2 is obvious b Show that E1 and E2 are Linvariant Find the matrices A1 LE1B1 and A2LE2B2 we have E1spanlefteginpmatrix2 1 1 1endpmatrixeginpmatrix1 1 1 1endpmatrixeginpmatrix1 1 2 1endpmatrix ight Let xin E1implies xeginpmatrix2alpha 1alpha 2alpha 3 alpha 1alpha 2alpha 3 alpha 1alpha See Answer

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