Section 6.4 Examples
\begin{equation*}
\Hom_F\big(U\bigoplus V,W\big)\simeq\Hom_F(U,W)\bigoplus\Hom_F(V,W).
\end{equation*}
Example 6.4.2. (Intersections and product).
\begin{equation*}
V\big/\bigcap_{i\in I}W_i\text{ is isomorphic to a subspace of }\prod_{i\in I}\big(V/W_i\big).
\end{equation*}
For each \(i\in I\) consider the natural projection \(\pi_i\colon V\to V/W_i\) (see Definition 6.2.1). Define \(\pi\colon V\to\prod_{i\in I}\big(V/W_i\big)\) by
\begin{equation*}
v\mapsto (\ldots,v+W_i,\ldots)
\end{equation*}
The verification that this map is \(F\)-linear is left to the reader. The kernel of \(\pi\) is
\begin{align*}
\ker(\pi)\amp=\{v\in V:v+W_i=0+W_i\text{ for each }i\in I\}\\
\amp=\{v\in V:v\in W_i\text{ for each }i\in I\}\\
\amp=\bigcap_{i\in I}W_i
\end{align*}
By Theorem 6.3.2, \(V\big/\bigcap_{i\in I}W_i\) is isomorphic to \(\Im(\pi)\text{.}\) Hence the result. This shows that the following is an exact sequence.
\begin{equation}
0\to \bigcap_{i\in I}W_i\to V\to\prod_{i\in I}\big(V/W_i\big).\tag{6.4.1}
\end{equation}
Example 6.4.3. (Intersection and Direct Sum).
\begin{equation*}
p\colon V/X\to V/(X+Y)\quad\text{and}\quad q\colon V/Y\to V/(X+Y)
\end{equation*}
Consider the compositions
\begin{equation*}
r\colon V/(X\cap Y)\to V/X\to (V/X)\bigoplus (V/Y)
\end{equation*}
and
\begin{equation*}
s\colon V/(X\cap Y)\to V/Y\to (V/X)\bigoplus (V/Y).
\end{equation*}
Let \(t\colon V/(X\cap Y)\to(V/X)\bigoplus (V/Y)\) be such that \(t=r+s\text{.}\) We show that
\begin{equation*}
p-q\colon (V/X)\bigoplus (V/Y)\to V/(X+Y)
\end{equation*}
is surjective, \(t\colon V/(X\cap Y)\to (V/X)\bigoplus (V/Y)\) is injective, and \(\Im(t)=\ker(p-q)\text{.}\) In other words, we show that the following is an short exact sequence.
\begin{equation}
0\to V/(X\cap Y)\xrightarrow{t}(V/X)\bigoplus (V/Y)\xrightarrow{p-q} V/(X+Y)\to 0\tag{6.4.2}
\end{equation}
Indeed, the injectivity is a part of Example 6.4.2. The surjectivity of \(p-q\) is left to the reader. We have
\begin{align*}
(p-q)\circ t\left(v+(X\cap Y)\right)\amp=(p-q)\left(v+X,v+Y\right)=0
\end{align*}
Hence, \(\Im(t)\leq\ker(p-q)\text{.}\) Now suppose that \((p-q)(v+X,v^\prime+Y)=0\text{,}\) i.e., \(v+(X+Y)=v^\prime+(X+Y)\text{.}\) Therefore there exists \(x\in X\) and \(y\in Y\) such that \(v-v^\prime=x+y\text{.}\) Hence,
\begin{equation*}
v-x=v^\prime+y.
\end{equation*}
Put \(w=v-x=v^\prime+y\text{.}\) Thus,
\begin{align*}
r\left(w+(X\cap Y)\right)\amp=r\left(v-x+(X\cap Y)\right)\\
\amp=\left((v-x)+X,0\right)\\
\amp=(v+X,0)
\end{align*}
Similarly,
\begin{align*}
s\left(w+(X\cap Y)\right)\amp=r\left(v^\prime-y+(X\cap Y)\right)\\
\amp=\left(0,(v^\prime+y)+Y\right)\\
\amp=(0,v^\prime+Y)
\end{align*}
In particular,
\begin{equation*}
(r+s)\left(w+(X\cap Y)\right)=t\left(w+(X\cap Y)\right)=(v+X,v^\prime+Y)\text{.}
\end{equation*}
Hence, \(\ker(p-q)\leq\Im(t)\) and the result is proved.
Example 6.4.4. (Intersection and Sum).
\begin{equation*}
W_1\big/(W_1\cap W_2)\simeq (W_1+W_2)/W_2.
\end{equation*}
