Complementary subspaces - is my textbook wrong? My textbook says that
"Let $V= R^3, W_1 = \{(\alpha,0,0)\}$ and $W_2 = \{(0,\alpha,\beta)\}$. Then $W_1 + W_2$ are complementary subspaces. 
However, the definition of a subspace is that $W_1 + W_2 = V$, but here $W_1+W_2 = \{(\alpha, \alpha, \beta) : \alpha, \beta \in R \}$, but that is limited points in $R^3$ where the $x$ and $y$ coordinates are the same. 
So is my textbook wrong or am I missing something?
 A: What you're missing is that the $\alpha$ in the definition of $W_1$ does not have to be the same as the $\alpha$ in the definition of $W_2$.  The definition of $W_1$ means that a vector is in $W_1$ if there exists $\alpha$ such that the vector is $(\alpha,0,0)$.  The definition of $W_2$ means that a vector is in $W_2$ if there exist $\alpha$ and $\beta$ such that the vector is $(0,\alpha,\beta)$.  In particular, the variable $\alpha$ is separately bound in each of these definitions, so there's no reason it has to be the same number in both of them if we are using both definitions at the same time.
For instance, $(1,0,0)\in W_1$ (since you could have $\alpha=1$).  Also, $(0,2,3)\in W_2$ (since you could have $\alpha=2$ and $\beta=3$).  Therefore the sum $(1,0,0)+(0,2,3)=(1,2,3)$ is an element of $W_1+W_2$.
A: The notation $\{(\alpha,0,0)\}$ is very informal. One more formal way to say the same is writing
$$\{(\alpha,0,0)\in\Bbb R^3:\alpha\in\Bbb R\},\quad \{(0,\alpha,\beta)\in\Bbb R^3:\alpha,\beta\in\Bbb R\}\tag{1}$$
And the sum of sets, in general, is defined as
$$A+B:=\{a+b:a\in A, b\in B\}\tag{2}$$
If we name
$$A:=\{(\alpha,0,0)\in\Bbb R^3:\alpha\in\Bbb R\},\quad B:=\{(0,\alpha,\beta)\in\Bbb R^3:\alpha,\beta\in\Bbb R\}$$
then observe that the definition in $(2)$ is not using the variables inside of the definitions of $A$ and $B$. Indeed because it is a definition the name of the variables that were used was arbitrary, I used the labels $A$, $B$, $a$ and $b$ just to represent the concept. 
In the same way the labels $\alpha$ and $\beta$ used in $(1)$ were arbitrary in each definition, that is, $\alpha,\beta$ represent arbitrary real numbers.
