Question

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?

Answer 1

$A:\\no: \\L\left( 0 \right) =\left[ 0+1,2\cdot 0 \right] =\left[ 1,0 \right] \ne 0\\B:\\yes:\\L\left( \alpha \left[ u_1,u_2 \right] +\beta \left[ v_1,v_2 \right] \right) =\\=\left[ 2\left( \alpha u_1+\beta v_1 \right) ,2\left( \alpha u_2+\beta v_2 \right) \right] =\alpha \left[ 2u_1,2u_2 \right] +\beta \left[ 2v_1,2v_2 \right] =\\=\alpha L\left( \left[ u_1,u_2 \right] \right) +\beta L\left( \left[ v_1,v_2 \right] \right)$