Determine if the transformation of V into given W is linear

Determine if the transformation of V into given W is linear

Deten nine si la | T:R\"R:TG) T:R2 R:T|- |=xy xy

Solution

Let x = (x₁,y₁)^T and y = (x₂,y₂)^T be 2 arbitrary vectors in R² and let k be an arbitrary scalar. Then T(x) = x₁y₁ and T(y) = x₂y₂ so that T(x)+T(y) = x₁y₁+ x₂y₂. Also T(x+y) = T((x₁,y₁)^T+(x₂,y₂)^T)= T(x₁+x₂, y₁+y₂)^T = (x₁+x₂)*(y₁+y₂) = x₁y₁+ x₂y₂+ x₁y₂ +x₂y₁. Thus, T(x+y) T(x)+T(y). Hence T does not preserve vector addition so that T is not a linear transformation.