I have this sentence in my notes:
A representation $D$ from a group $G$ to matrices $M(V)$ acting on a vector space $V$ is irreducible if and only if $V$ has no non-trivial invariant proper subspaces.
I have the following questions:
Why do we write $M(V)$, not just $M$? Does this mean that the matrices $M(V)$ is referring to are made of columns of vectors in vector space $V$? That would be reasonable I believe.
Does "no non-trivial invariant proper subspaces" mean that given any vector $v \in V $ and and matrix $M \in D$, and if $M$ acts on $v$, than all of its components change? In other words, none of the components of $Mv$ and $v$ are the same. Is this correct?
As I understand it a representation of $G$ on $V$ would be a homomorphism, say $D$, from $G$ to $M(V)$.
The notion of an invariant subspace is then, that of a subspace $U\subset V$ with $AU\subset U\,\forall A\in D(G)$.