Let U and W be subspaces of a vector space V such that W ⊆ U. Prove that U/W is a subspace of V/W and that (V/W)/(U/W) is isomorphic to V/U The book says to do it by defining a function T:V/W-V/U by the rule T(v+W) = v+U. Show that T is a well defined linear transformation and applying 1st isomorphism thm (V/Ker(T) iso to Im(T))

Question

Let U and W be subspaces of a vector space V such that W ⊆ U. Prove that U/W is a subspace of V/W and that (V/W)/(U/W) is isomorphic to V/U The book says to do it by defining a function T:V/W->V/U by the rule T(v+W) = v+U. Show that T is a well defined linear transformation and applying 1st isomorphism thm (V/Ker(T) iso to Im(T))

watchmath · Answer

I believe you can solve $U/W$ is a subspace of $V/W$ by yourself. To use the 1st isomorphism theorem, we just need to show that the $T$ that you defined above is onto $V/U$ and $\ker T=U/W$.
Let $x+U\in V/U$, then by definition $T(x+W)=x+U$. Hence $T$ is onto.
If $x+W \in U/W$ then $x\in U$. It follows that $T(x+W)=x+U=0$ (since $x\in U$). So $U/W\subset \ker T$. Conversely let $x+W \in \ker T$. Then $T(x)=x+U=0$. Hence $x\in U$. Thus $x+W\in U/W$. So $\ker T\subset U/W$.
Therefore $\ker T= U/W$.