A. Let L: R3 R3 be defined by L([u₁ u₂ u3]) = [u₁ + 1, 2u₂]. Is L a linear transformation?
B. Let L: R₂ → R₂ be defined by L([u₁ u₂]) = [2u1 2u₂]. Is L a linear transformation?
A:no:L(0)=[0+1,2⋅0]=[1,0]≠0B:yes:L(α[u1,u2]+β[v1,v2])==[2(αu1+βv1),2(αu2+βv2)]=α[2u1,2u2]+β[2v1,2v2]==αL([u1,u2])+βL([v1,v2])A:\\no: \\L\left( 0 \right) =\left[ 0+1,2\cdot 0 \right] =\left[ 1,0 \right] \ne 0\\B:\\yes:\\L\left( \alpha \left[ u_1,u_2 \right] +\beta \left[ v_1,v_2 \right] \right) =\\=\left[ 2\left( \alpha u_1+\beta v_1 \right) ,2\left( \alpha u_2+\beta v_2 \right) \right] =\alpha \left[ 2u_1,2u_2 \right] +\beta \left[ 2v_1,2v_2 \right] =\\=\alpha L\left( \left[ u_1,u_2 \right] \right) +\beta L\left( \left[ v_1,v_2 \right] \right)A:no:L(0)=[0+1,2⋅0]=[1,0]=0B:yes:L(α[u1,u2]+β[v1,v2])==[2(αu1+βv1),2(αu2+βv2)]=α[2u1,2u2]+β[2v1,2v2]==αL([u1,u2])+βL([v1,v2])
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