Exercises 4.10 Exercises
We gather some easy to derive properties of exact sequences. We assume that \(X,Y, Z,\) and \(W\) are vector spaces (not necessarily finite-dimensional) over a field \(F\text{.}\)
We make the following conventions.
- The trivial subspace \(\{0\}\) of a vector space will be denoted by \(0\) when used in a diagram.
- Since there is a unique linear map from the trivial subspace to the vector space (resp., from a vector space to its trivial subspace) we will not write the name for this linear map.
2.
Show that \(0\xrightarrow{}X\xrightarrow{f}Y\) is exact if and only if \(f\) is injective.3.
Show that \(Y\xrightarrow{g}Z\xrightarrow{} 0\) is exact if and only if \(g\) is surjective.4.
Show that \(0\xrightarrow{}X\xrightarrow{f}Y\xrightarrow{}0\) is exact if and only if \(f\) is bijective if and only if \(f\) is an \(F\)-isomorphism.5.
Show that given an \(F\)-linear map \(f\colon X\to Y\) we have the following short exact sequence.
\begin{equation*}
0\to\ker(f)\xrightarrow{}X\xrightarrow{f}\Im(f)\xrightarrow{}0.
\end{equation*}
6.
Let \(P\) be a finite-dimensional vector space over a field \(F\text{.}\) Suppose that we have the following diagram with exact row and \(g\colon P\to Y\) is an \(F\)-linear map:
\begin{equation*}
\begin{CD}@.P@.\\@. @VgVV @.\\X@>f>>Y@>>>0\end{CD}
\end{equation*}
Show that there exists an \(F\)-linear transformation \(j\colon P\to X\) such that \(f\circ j=g\text{.}\)
Hint.
Let \(\{p_1,p_2,\ldots,p_n\}\) be a basis for \(P\) over \(F\text{.}\) Since, \(f\) is surjective (see Exercise 4.10.3), there exists \(x_i\in X\) such that \(f(x_i)=g(p_i)\) for each \(i\text{.}\) Define \(j(\sum\alpha_ip_i)=\sum\alpha_i x_i\text{.}\)
7.
Let \(Y\) be a finite-dimensional vector space and \(I\) be a vector space over \(F\text{.}\) Suppose that we have the following diagram with exact row and \(\ell\colon X\to I\) is an \(F\)-linear map:
\begin{equation*}
\begin{CD}0@>>> X@>i>>Y\\@.@V\ell VV@.\\@.I@.\end{CD}
\end{equation*}
Show that there exists an \(F\)-linear map \(d\colon Y\to I\) such that \(d\circ i=\ell\text{.}\)
Hint.
By Exercise 4.10.2, \(i\) is injective and hence \(X\) is also finite-dimensional. Let \(\{x_1,x_2,\ldots,x_n\}\) be a basis of \(X\text{.}\) Then, \(\{i(x_1),i(x_2),\ldots,i(x_n)\}\) is an \(F\)-linearly independent subset of \(Y\text{.}\) Extend \(\{i(x_1),i(x_2),\ldots,i(x_n)\}\) to a basis for \(Y\text{,}\) say \(\{i(x_1),i(x_2),\ldots,i(x_n),y_1,\ldots,y_m\}\text{.}\) Put \(d(i(x_i))=\ell(x_i)\) and \(d(y_i)=0\text{.}\)
