I am doing some revision for upcoming exams and as a practice I am trying to find an example of a Group Homomorphism for the following domains and codomains:
A) $GL_4(\mathbb{R}) \to GL_1(\mathbb{R})$
B) $GL_3(\mathbb{R}) \to GL_4(\mathbb{R})$
C) $GL_3(\mathbb{R}) \to SL_3(\mathbb{R})$
D) $GL_3(\mathbb{R}) \to GL_2(\mathbb{R})$
I know the answer to A) is the determinant, and I'm pretty sure that part d) isn't possible, but could someone please help me with the other two?
Thanks!
On
For B), get intuition by moving from $2$-space to $3$-space; if we have a two-dimensional rotation matrix $\pmatrix{a & b \\ -b & a}$, we can do the same operation but in $3$ dimenions by fixing the $z$-axis. The map
$$M = \pmatrix{a & b \\ -b & a} \mapsto \pmatrix{a & b & 0 \\ -b & a &0 \\ 0 &0 &1} = \pmatrix{M & 0 \\ 0 & 1} \in GL_3(\Bbb R)$$ works, and will work for general transformations (not just rotations) to hop up as many dimensions as you like.
You can do something similar in D), by transforming in $3$-space then projecting onto the $xy$-plane and ignoring what happens to the $z$-coordinate to go from $GL_3(\Bbb R)$ to $GL_2(\Bbb R)$.
The canonical map in C) is the map that sends a matrix $M$ to $\frac{1}{\det M}M$.
I'm going to cheat a bit and generalise the answer for part A.
For parts A, B, and D, each matrix $M$ maps to a diagonal matrix where all diagonal entries are the determinant of $M$.
For part C, offhand I think you can take each matrix $M$ mapping to a scaled version of $M$, that is, $kM$ where $k\in\mathbb{R}$, so that $\det (kM) = 1$.