## anonymous 5 years ago 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))

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 $$\ker T= U/W$$.