let R* be the group of nonzero real numbers under multiplication. then the deter

ID: 1941526 • Letter: L

Question

let R* be the group of nonzero real numbers under multiplication. then the determinant mapping A-> det(A) is a homomorphism from GL(2,R) to R*. the kernel of the determinant mapping is SL(2,R). i am suppose to show that this is a homomorphism but i have no idea where to go and what to do

Explanation / Answer

We have a map f:GL(2,R) ----> R* defined by f(A)=det(A). Homomorphism: we have to show that f(AB)=f(A)f(B) for any A,B in GL(2,R). To see this, we have to use only known property, namely, det(AB)=det(A).det(B). So, f(AB)=det(AB)=det(A)det(B)=f(A)f(B). So f is an homomorphism. Kerf:={A in GL(2,R): f(A)=det(A)=1}, where 1 is the unity in R*. But from definition SL(2,R)={A in GL(2,R): det(A)=1}. So kerf=Sl(2,R). Hence the proof.

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let R* be the group of nonzero real numbers under multiplication. then the deter

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