Com efeito, dado \(\mathbf{A} =
\left[
\begin{array}{rr}
a \amp b \\
c \amp d
\end{array}
\right]
\in \mathcal M_{2\times 2}\text{,}\) então
\begin{align*}
\mathbf{A} \amp=
\left[
\begin{array}{rr}
a \amp b \\
c \amp d
\end{array}
\right]
=
\left[
\begin{array}{rr}
a \amp b \\
0 \amp 0
\end{array}
\right]
+
\left[
\begin{array}{rr}
0 \amp 0 \\
c \amp d
\end{array}
\right]
= \mathbf{A}_1 + \mathbf{A}_2
\end{align*}
onde \(\mathbf{A}_1 \in U\) e \(\mathbf{A}_2 \in W\text{.}\) Logo, \(V = U + W\text{.}\) Além disso,
\begin{align*}
U \cap W \amp=
\left\{
\left[
\begin{array}{cc}
a \amp b \\
0 \amp 0
\end{array}
\right] : a,b \in \mathbb R
\right\}
\cap
\left\{
\left[
\begin{array}{cc}
0 \amp 0 \\
c \amp d
\end{array}
\right] : c,d \in \mathbb R
\right\}
=
\left\{
\left[
\begin{array}{cc}
0 \amp 0 \\
0 \amp 0
\end{array}
\right]
\right\}
\end{align*}
Como
\(U \cap W = \{\mathbf{0}\}\text{,}\) resulta do
Teorema 4.3.1 que
\(V = U \oplus W\text{.}\)
