I am reading "Linear Algebra Done Right 3rd Edition" by Sheldon Axler.
3.69
Suppose $V$ is finite-dimensional and $T\in\mathcal{L}(V)$. Then the following are equivalent:
(a) $T$ is invertible;
(b) $T$ is injective;
(c) $T$ is surjective.
3.D.9:
Suppose $V$ is finite-dimensional and $S,T\in\mathcal{L}(V)$. Prove that $ST$ is invertible if and only if both $S$ and $T$ are invertible.
My solution:
If both $S$ and $T$ are invertible, then both $S$ and $T$ are injective by 3.69 on p.87.
Obviously, $ST$ is also injective.
So, $ST$ is invertible by 3.69 on p.87.If $ST$ is invertible, then $ST$ is injective and surjective by 3.69 on p.87.
If $T$ is not injective, then obviously, $ST$ is also not injective.
So, $T$ must be injective.
So, $T$ is invertible by 3.69 on p.87.
Since $ST$ is surjective, there exists $v\in V$ such that $(ST)v=w$ for any $w\in V$.
Since $(ST)v=S(Tv)=w$, $S$ is surjective.
So, $S$ is invertible by 3.69 on p.87.
3.D.13
Suppose $V$ is a finite-dimensional vector space and $R,S,T\in\mathcal{L}(V)$ are such that $RST$ is surjective. Prove that $S$ is injective.
My solution:
Since $RST$ is surjective, $RST$ is invertible by 3.69 on p.87.
So, by 3.D.9, $R$, $S$ and $T$ are invertible.
So, $S$ is injective by 3.69 on p.87.
The following $12$ statements are all true.
I wonder why the author chose the following exercise:
Suppose $V$ is a finite-dimensional vector space and $R,S,T\in\mathcal{L}(V)$ are such that $RST$ is surjective. Prove that $S$ is injective.
Any reason?
Or did the author choose randomly?
Suppose $V$ is a finite-dimensional vector space and $R,S,T\in\mathcal{L}(V)$ are such that $RST$ is surjective. Prove that $R$ is injective.
Suppose $V$ is a finite-dimensional vector space and $R,S,T\in\mathcal{L}(V)$ are such that $RST$ is surjective. Prove that $S$ is injective.
Suppose $V$ is a finite-dimensional vector space and $R,S,T\in\mathcal{L}(V)$ are such that $RST$ is surjective. Prove that $T$ is injective.
Suppose $V$ is a finite-dimensional vector space and $R,S,T\in\mathcal{L}(V)$ are such that $RST$ is injective. Prove that $R$ is surjective.
Suppose $V$ is a finite-dimensional vector space and $R,S,T\in\mathcal{L}(V)$ are such that $RST$ is injective. Prove that $S$ is surjective.
Suppose $V$ is a finite-dimensional vector space and $R,S,T\in\mathcal{L}(V)$ are such that $RST$ is injective. Prove that $T$ is surjective.
Suppose $V$ is a finite-dimensional vector space and $R,S,T\in\mathcal{L}(V)$ are such that $RST$ is surjective. Prove that $R$ is surjective.
Suppose $V$ is a finite-dimensional vector space and $R,S,T\in\mathcal{L}(V)$ are such that $RST$ is surjective. Prove that $S$ is surjective.
Suppose $V$ is a finite-dimensional vector space and $R,S,T\in\mathcal{L}(V)$ are such that $RST$ is surjective. Prove that $T$ is surjective.
Suppose $V$ is a finite-dimensional vector space and $R,S,T\in\mathcal{L}(V)$ are such that $RST$ is injective. Prove that $R$ is injective.
Suppose $V$ is a finite-dimensional vector space and $R,S,T\in\mathcal{L}(V)$ are such that $RST$ is injective. Prove that $S$ is injective.
Suppose $V$ is a finite-dimensional vector space and $R,S,T\in\mathcal{L}(V)$ are such that $RST$ is injective. Prove that $T$ is injective.
Best Answer
Can’t comment yet so will just answer. I think you are correct it was just chosen randomly more or less. My guess is the goal is for you to see that if the product has nice properties (invertible or the equivalent) then so do the individual matrices.