Let $1\leq n\in \mathbb{N}$, $V:=\mathbb{R}^n$, $M:=(V^V, \circ)$ and $\text{End}_{\mathbb{R}}(V):=\{\phi :V\rightarrow V\mid \phi \text{ is linear }\}$.
Show the following:
- $\text{End}_{\mathbb{R}}(V)$ is closed under $\circ$.
- If $f,g\in V^V$ such that $f\circ g=\text{id}_V$, then $f$ is surjective and $g$ is injective.
- If $\phi, \psi \in \text{End}_{\mathbb{R}}(V)$ such that $\phi \circ \psi =\text{id}_V$, then $\phi$ and $\psi$ are invertible and it holds that $\phi^{-1}=\psi$.
- Let $a,b\in M_n(\mathbb{R})$ such that $ab=u_n$. Then $a$ and $b$ are invertible and it holds that $a^{-1}=b$. Hint: Consider $\phi_a$.
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I have done the following:
As for 1:
Let $\phi, \psi \in \text{End}_{\mathbb{R}}(V)$. Then since $\phi :V\rightarrow V$ and $\psi :V\rightarrow V$ it follows that $\phi\circ \psi :V\rightarrow V$. Since $\phi$ is linear we have that $\phi (ax+y)=a\phi (x)+\phi (y )$ for $x,y\in V$. Equivalently, since $\psi$ is linear we have that $\psi (b\tilde{x}+\tilde{y})=b\psi (\tilde{x})+\psi (\tilde{y})$ for $\tilde{x},\tilde{y}\in V$. We have that for $\tilde{x},\tilde{y}\in V$ that $$\phi \circ \psi (b\tilde{x}+\tilde{y})=\phi (\psi (b\tilde{x}+\tilde{y}))=\phi (b\psi (\tilde{x})+\psi (\tilde{y}))=b\phi (\psi (\tilde{x}))+\phi (\psi (\tilde{y}))=b\phi \circ \psi (\tilde{x})+\phi \circ \psi (\tilde{y})$$ Therefore $\phi\circ \psi $ is linear and so $\phi\circ \psi\in \text{End}_{\mathbb{R}}(V)$ which means that $\text{End}_{\mathbb{R}}(V)$ is closed under $\circ$.
Is everything correct?
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As for 2:
Let $g(a)=g(b)$. Then we have that $$f(g(a))=f(g(b)) \Rightarrow (f\circ g)(a)=(f\circ g)(b) \Rightarrow \text{id}_V(a)=\text{id}_V(b) \Rightarrow a=b$$ This means that $g$ is injective.
Let $y\in V$. Let $x=g(y )$ (we can set this like that, or not? ). Then we get $f(x)=f(g(y )) \Rightarrow f(x)=(f\circ g)(y ) \Rightarrow f(x)=\text{id}_V(y ) \Rightarrow f(x)=y$. So for every $y \in V$ , there exists a $x \in X$ such that $f(x) = y$, which means that $f$ is surjective.
Is everything correct?
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As for 3:
Can we use here statement 2 ?
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As for 4:
What exactly is $\phi_a$ ?