From the question,
We'll show that f is linear if for A1,A2∈R2∗2,α,β∈R, we have that f(αA1+βA2)=αf(A1)+βf(A2) To this end,
Let A1=(acbd),A2=(egfh)So,αA1+βA2=α(acbd)+β(egfh)⟹αA1+βA2=(αa+βeαc+βgαb+βfαa+βh)Now,f(αA1+βA2)=(αA1+βA2)vf(αA1+βA2)=(αa+βeαc+βgαb+βfαa+βh)(−12)f(αA1+βA2)=(−αa−βe+2αb+2βf−αc−βg+2αa+2βh)f(αA1+βA2)=(−αa+2αbαc+2αd)+(−βe+2βf−βg+2βh)f(αA1+βA2)=α(acbd)(−12)+β(egfh)(−12)f(αA1+βA2)=αA1v+βA2vf(αA1+βA2)=αf(A1)+βf(A2)Thus, f is a linear mapping
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