I started reviewing linear algebra, from a different textbook (Axler's), after taking a fast paced summer class. Unfortunately, I've become confused with a concept that is introduced at the end of chapter one. That is, sum of subspaces.
Axler's text defines the sum of subspaces as follows.
Let $U_1,U_2,…,U_m$ be subspaces of a vectorspace $V$. Then we say $U_1+U_2+…+U_m=\{u_1+…+u_m:u_1\in U_1,…,u_m\in U_m\}$
I thought I understood this concept, but I'm afraid I don't because I am having trouble answering the following assertions he asks us to verify.
First that if we let $U_1,U_2,…,U_m$ be subspaces of a vectorspace $V$, then the sum of those subspaces is a subspace of $V$.
Also, this is what really had me tricked to thinking I understood it. Let $U=\{(x,0,0)\in \mathbb R^3: x\in \mathbb R\}$ and $W=\{(0,y,0)\in \mathbb R^3:y\in \mathbb R\}$, then $U+W= \{(x,y,0):x,y\in \mathbb R\}$. So this example made me think it was pretty straight forward and that I understood it, but in the next few lines he says let $Z= \{(y,y,0)\in \mathbb R^3:y\in\mathbb R\}$. Then $U+W=U+Z$ (which I am asked to verify).
Could someone please help me understand the definition and the verifications?
EDIT: I currently see the definition to say that when we take a collection of sets that are subspaces the sum of the sets is a set which consists of the sum of all their elements. However when I say that it seems to me the summed set consists of just one elements (the total sum of all the elements).
Thank You
Best Answer
A point that might confuse you is that the same letters are used, let us rewrite this.
If you have some element from $U+W$ then it is of the form $(x,y,0)$ for some $x,y$.
If you have some element from $U+Z$ then it is of the form $(v+w,w,0)$ for some $v,w$.
Now the claim is that these two really yield the same collection of elements. I assume you can see that each element of the latter form is of the former. More specifically if we pick a generic element $ (v+w, w,0) $ in $U+Z$, we can equivalently express the element in the form $ (x,y,0) $ by setting $ x = v + w$ and $ y = w $. This shows that $ U + Z \subset U + W$.
For the other direction we can select an arbitrary element $(x, y, 0) $ of $ U + W $ and re-express it in the form $(v+w,w,0)$ by setting $ v = x - y $ and $ w = y $. The latter shows $ U + W \subset U + Z$.
The two subset relations $U+Z \subset U+W $ and $ U + W \subset U + Z$ imply that $U+Z = U+W$.
Added following the edit: a single element of the sum $U+W$ is one element of $U$ plus one element of $W$. And $U+W$ is the set of all these elements together.
So, you have since $(19,0,0)$ in $U$ and $(0,-3,0)$ in $W$ that $(19,0,0)+(0,-3,0)$ in $U+W$. However, you can of course evaluate that sum $(19,0,0)+(0,-3,0)= (19,-3, 0)$.