For the V vector space contains all 2×2 matrices. Determine whether the T:V→R1 is the linear transformation over the T(A)=a + 2b − c + d, where A=[abcd]

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For the V vector space contains all 2×2 matrices. Determine whether the T:V→R1 is the linear transformation over the T(A)=a + 2b − c + d, where A=[abcd]

berggansS · Accepted Answer

Calculation: For the vector space V, that contains all 2×2 matrices Consider two matrices be lie in that vector space V be, A=[abcd]B=[pqrs] and  Such that T(A)=a + 2b − c + d T(B)=p + 2q − r + s Perform additional operation. A + B=[abcd]+[pqrs] =[a+p+b+qc+r+d+s] And, T(A) + T(B)=(a + 2b − c + d) + (p + 2q − r + s) =a + p + 2b + 2q − c − r + d + s =(a + p) + 2(b + q) − (c + r) + (d + s) Also, T(A + B)=(a + p) + 2(b + q) − (c + r) + (d + s) Thus it shows that T(A+B)=T(A)+T(B), that means it satisfies the addition property Consider the condition of T(kA)=ka + 2kb − kc + kd =k(a + 2b − c + d) =kT(A) So the following expression also satisfies the scalar property. Answer: As for the vector space V, the property of addition and scalar are satisfied. Thus the expression of T(A)=a+2b−c+d is the linear transformation

For the V vector space contains all 2 times 2 matrices.

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