Prove that the given transformation is a linear transformation, using the definition. -Y x + 8y 6x - 7y T[X] - Definition: A transformation T: R → R is called a linear transformation if the following is true. 1. T(u + v) = T(u) + T(v) for all u and v in Rº 2. T(cv) = CT(v) for all v in R and all scalars c Use the following remark. Remark: The definition of a linear transformation can be streamlined by combining (1) and (2) as shown below. T(C₁ V₁ + C₂V₂) = C₁ T (V₁) + C₂ T(V₂) for all v₁, v₂ in R" and scalars C₁, C₂ 17 T is a linear transformation if and only if T(C₁ V₁ + C₂ V2₂) = C₁ T(V₁) + C₂ T(V₂), where v₁ = Then we get the following. T(C₁V₁ + C₂V₂) Thus T is linear. = 1(+[;]++[]) = T -1 -(C₁Y1 (C₁₂X₁ + ₂x₂) + ( _) (₁₂ x ₂ + €₂X₂) + (1 -C11 C₁x₂ + (1 ))²₁x₂ + (1 + C₂Y2) C₁ T(V₁) + C₂T(V₂) 11 ) (C₁Y/₁ X1 €₂V/₂ H -Y1 LCH x₁ + ( X1 C1 ))x₂ + ( V₂ + C₂Y2) (²₂₁ +²₂₂) = x2 -C212 €₂x₂ + ( _)<₂*₂ + (1 x₂ + ( x2 -Y2 _)x₂ + (1 Y₂ mal

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