Currently reading through Axler's Linear Algebra Done Right 3rd ed. I am on the definition of direct sums and just want to clarify the notation.
It says:
Suppose $U_1, …, U_m$ are subspaces of V. Every element of $U_1 + \cdots + U_2$ can be written in the form
$u_1 + \cdots + u_m$
where each $u_j$ is in $U_j$.
Example:
Suppose U is the subspace of $F^3$ of those vectors whose last two coordinates equals $0$, and W is the subspace of $F^3$ of those vectors whose first two coordinates equals $0$:
$U=\{(x,y,0)\in F^3 : x,y \in F \}$ and $W= \{(0,0,z)\in F^3:z \in F \}$
Then $F^3 = U \oplus W$
So what I'm confused about is the notation. The elements of $U$ are $(x,y,0)$. Would these three elements be considered as $u_1$ for $U_1$, and then W's elements $(0,0,z)$ would be considered as $u_2$ for $U_2$? So then $u_1 + u_2 = (x,y,0) + (0,0,z) = (x,y,z)$?
Best Answer
I assume you mean "$U$ is the subspace whose last coordinate equals $0$". In that case, you're right, the example has $m=2$, $U_1=U$ and $U_2=W$, and in this case we have a direct sum $F^3=U_1\oplus U_2=U\oplus W$, so an arbitrary element $u=(x,y,z)\in F^3$ has a unique decomposition \begin{align} u&=(x,y,z)=\underbrace{(x,y,0)}_{:=u_1}+\underbrace{(0,0,z)}_{:=u_2}\in U_1+U_2 = U+W. \end{align} (It is due to the uniqueness of the decomposition that we put the extra decorative symbol $\oplus$ to indicate the "directness" of the sum $F^3=U\oplus W$).