I am reading the textbook Linear Algebra Done Right Chapter 3 section E on Products and Quotients of Vectors Spaces.
The goal is to make $V/U$ into a vector space. So, my understanding is that we need to define addition and scalar multiplication of the quotient space, and show that it is a vector space.
I can understand the definition of addition and scalar multiplication on $V/U$:
$$(v+U) + (w+U) = (v+w)+U$$
$$\lambda (v+U) = (\lambda v) + U$$
But the author said to do this (have the above definition), we need the following result.
Suppose $U$ is a subspace of $V$ and $v,w \in V$. Then the following are equivalent:
$$v-w \in U$$
$$v+U=w+U$$
$$(v+U) \cap (w+U) \neq \emptyset$$
Why the above result is needed to definethe addition and scalar multiplication?
Best Answer
Further to J.W. Tanner's comment, let us consider an example which might help to illustrate why the above result is necessary. Specifically, why it allows for a well-defined notion of addition in $V/U.$ That is, a definition of addition on $V/U$ that 'makes sense.'
Let us take the situation in Example 3.80 in Axler as our example. That is, let $U = \left\{ (x, 2x) \in \mathbb R^2 | x \in \mathbb R \right\}$. Then $U$ is the line through the origin in $\mathbb R^2$ with slope $2$. Thus $(17,20)+U$ is the line in $\mathbb R^2$ that contains the point $(17,20)$ and has slope $2$. By definition, $(17,20)+U$ is an affine subset that is parallel to $U$. Let us take another point on this line, say $(7,0)$ and consider the element $(7,0) + U$. Since $(7,0)$ is a point on the same line, we see that $(17,20)+U$ and $(7,0)+U$ represent the same element of $V/U$.
Now suppose we were to add each of these affine subsets to some other affine subset, say $(3,10)+U$, using the definition of addition on $V/U$. Then we get $((17,20)+U) + ((3,10)+U) = (20,30)+U$ in the first case, and $((7,0)+U)+((3,10)+U) = (10,10)+U$ in the second case. The question is, how do we know that $(20,30)+U$ and $(10,10)+U$ represent the same element? Well, using the first equivalence in the result, we see that $(20,30)-(10,10)= (10,20) \in U,$ which implies that $(20,30)+U = (10,10)+U$. That is, they do in fact represent the same element. So hopefully this goes to show why the above result contributes to a well-defined notion of addition in $V/U$. A similar story holds for scalar multiplication.
Another thing you might like to do is carefully read the proof of 3.87 (Quotient Space is a Vector Space). The above result is critical in proving that the provided definitions of addition and scalar multiplication make sense.