Using the First Isomorphism Theorem to find a linear transformation

Question

For a vector space V with subspaces U and W, how can we find a suitable linear transformation that by the first isomorphism theorem produces (U + W)/W $\simeq$ U/(U $\displaystyle \cap$ W)?

I gather that we will want U and W be a direct sum since the first isomorphism theorem gives us V/kerT $\simeq$ im T, i.e., V/U $\simeq$ W for a transformation T on W along U, but I am struggling to find a suitable linear transformation that would give (U + W)/W $\simeq$ U/(U $\displaystyle \cap$ W).

Answer 1

No, $U$ and $W$ will not form a direct sum, that's sort of the point of this theorem. There are two options here:

1. find a surjective linear map $U+W \to U/(U\cap W)$ whose kernel is $W$, or;
2. find a surjective linear map $U \to (U+W)/W$ whose kernel is $U\cap W$.

It turns out that option 2 is the most natural one: you take an element $u\in U$, observe that it is in $U+W$ as well (since $0\in W$ and $u=u+0$), and then take the class $u+W$, which is in $(U+W)/W$.

Why is this surjective? Any element of $(U+W)/W$ has the form $(u+w)+W$ for some $u\in U$ and $w\in W$. Then $(u+w)+W = u+W$ is the image of $u$.

Why is the kernel equal to $U\cap W$? Because an element $u+W$ is the zero vector of $(U+W)/W$ is and only if $u\in W$.

This is essentially going over the definitions carefully.