### final edits of map2

parent 6b7148f8
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 ... ... @@ -12012,7 +12012,7 @@ \end{equation*} We need only to verify that this map is onto. Any member of $W$ can be written as a linear combination of We can write any member of $W$ as a linear combination of basis elements $c_1\cdot \vec{w}_1+\dots+c_k\cdot \vec{w}_k$. This vector is the image, under the map described above, of ... ... @@ -12148,7 +12148,7 @@ $$r=h(c\vec{v})$$. Thus the rank of $$h$$ is one. Because the nullity is given as $n$, the dimension of the domain of Because the nullity is $n$, the dimension of the domain of $$h$$, the vector space $$V$$, is $$n+1$$. We can finish by showing $$\set{\vec{v},\vec{\beta}_1,\dots,\vec{\beta}_n}$$ ... ... @@ -12193,7 +12193,7 @@ c_1\vec{\beta}_1+\dots+c_n\vec{\beta}_n \mapsunder{h} c_1x_1+\dots+c_nx_n \end{equation*} is easily seen to be linear, and to be mapped by $\Phi$ to the is linear and $\Phi$ maps it to the given vector in $\Re^n$, so $$\Phi$$ is onto. The map $$\Phi$$ also preserves structure:~where ... ... @@ -12240,12 +12240,12 @@ \begin{ans}{Three.II.2.43} Either yes (trivially) or no (nearly trivially). If $$V$$ is homomorphic to' $$W$$ is taken to mean there is a If we take $$V$$ is homomorphic to' $$W$$ to mean there is a homomorphism from $$V$$ into (but not necessarily onto) $$W$$, then every space is homomorphic to every other space as a zero map always exists. If $$V$$ is homomorphic to' $$W$$ is taken to mean there is an If we take $$V$$ is homomorphic to' $$W$$ to mean there is an onto homomorphism from $$V$$ to $$W$$ then the relation is not an equivalence. For instance, there is an onto
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