Showing that a matrix is a homomorphism

[flomayoc]

Hi!
I am currently working on this question about matrices and showing they are homomorphisms. I have done part (i), but on part (ii) I am confused as the matrix is mapping to a - I have never seen this before and I'm not sure how to approach it. I know that usually you would work out the determinant and show that det(AB)=det(A)det(B), but I can't figure out how this would be different.

View attachment 7665

Thanks in advance!

 

[Euge]

Welcome, flomayoc! (Wave)

To show that $f$ is a homomorphism, take arbitrary elements $x,y\in T$, and prove $f(xy) = f(x)f(y)$. If $x,y\in T$, then $x = \begin{pmatrix}a & b\\0 & a^{-1}\end{pmatrix}$ and $y = \begin{pmatrix}c & d\\0 & c^{-1}\end{pmatrix}$ for some $a,c\in \Bbb R^*$ and $b,d\in \Bbb R$. Now what is the matrix product $xy$?.

 

[Deveno]

For part (iii), show for any $a \in \Bbb R^{\ast}$, that there exists $A \in T$ with $f(A) = a$. This shows $f$ is surjective, and then you can use the Fundamental Isomorphism Theorem.