MHB Prove G is a Subspace of V ⊕ V and Quotient Space (V ⊕ V)/G Isomorphic to V

[toni07]

Prove that G is a subspace of V ⊕ V and the quotient space (V ⊕ V) / G is isomorphic to V.

Let $V$ be a vector space over $\Bbb{F}$, and let $T : V \rightarrow V$ be a linear operator on $V$. Let $G$ be the subset of $V \oplus V$ consisting of all ordered pairs $(x, T(x))$ for $x$ in $V$. I don't really understand what I am supposed to do. Is $G$ a subset of $V \oplus V$ or a subset of the ordered pairs $(x, T(x))$. Please help.

 

[Opalg]

Re: Prove that G is a subspace of V ⊕ V and the quotient space (V ⊕ V) / G is isomorphic to V.

Let $V$ be a vector space over $\Bbb{F}$, and let $T : V \rightarrow V$ be a linear operator on $V$. Let $G$ be the subset of $V \oplus V$ consisting of all ordered pairs $(x, T(x))$ for $x$ in $V$. I don't really understand what I am supposed to do. Is $G$ a subset of $V \oplus V$ or a subset of the ordered pairs $(x, T(x))$. Please help.

By definition, $V \oplus V$ consists of all ordered pairs $(x,y)$ for $x$ and $y$ in $V$.

$G$ consists of those elements $(x,y)$ in $V \oplus V$ for which $y = Tx$. So $G$ is a subset of $V \oplus V$, and your first task is to show that it is a subspace.

 

[Chris L T521]

Re: Prove that G is a subspace of V ⊕ V and the quotient space (V ⊕ V) / G is isomorphic to V.

Let $V$ be a vector space over $\Bbb{F}$, and let $T : V \rightarrow V$ be a linear operator on $V$. Let $G$ be the subset of $V \oplus V$ consisting of all ordered pairs $(x, T(x))$ for $x$ in $V$. I don't really understand what I am supposed to do. Is $G$ a subset of $V \oplus V$ or a subset of the ordered pairs $(x, T(x))$. Please help.

Well you're told that elements of $G$ consist of all ordered pairs $(x,T(x))$ for $x\in V$; i.e.

\[G=\{(x,T(x)):x\in V\}\subset V\oplus V\]

To show that $G$ is a subspace of $v$, you want to take $u=(x_1,T(x_1))$ and $v=(x_2,T(x_2))$ and show that for any $a,b\in \Bbb{F}$, you have that $au+bv \in G$ (this is rather easy to show because addition is done component wise; furthermore you also know that $T$ is linear).

This is me going off on a limb (kind of; so I'd appreciate it if someone could point out if I'm wrong here), but I believe that in order to show that $(V\oplus V) / G\cong V$, you need to construct a morphism $\varphi : V\oplus V \rightarrow V$ where $\mathrm{Im}(\varphi)=V$ and $\ker\varphi = G$ (and thus you obtain the result by the First Isomorphism Theorem).