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))

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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))

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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 \(\ker T= U/W\).

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