the function det: M 2×2 (F)–> F is a linear transformation
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Yes, the function det: M₂ₓ₂(F) → F is a linear transformation.
To show that a function is a linear transformation, we must prove that it satisfies two properties: (1) additivity and (2) homogeneity.
1. Additivity: Let A, B ∈ M₂ₓ₂(F). Then, det(A + B) = (a₁₁ + b₁₁)(a₂₂ + b₂₂) – (a₁₂ + b₁₂)(a₂₁ + b₂₁). By expanding the terms, we get det(A + B) = det(A) + det(B). Thus, det satisfies additivity property.
2. Homogeneity: Let A ∈ M₂ₓ₂(F) and k ∈ F. Then, det(kA) = (ka₁₁)(ka₂₂) – (ka₁₂)(ka₂₁) = k²(a₁₁a₂₂ – a₁₂a₂₁) = k²det(A). Thus, det satisfies homogeneity property.
Since det satisfies both additivity and homogeneity properties, it is a linear transformation.
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