View source for Notes for AKT-140305/0:41:57

'''Showing that''' <math>(\ker f)^* = V^*/\mathrm{im}f^*</math> '''for a linear map''' <math>f : V \rightarrow W, V\; \mathrm{and}\; W</math> '''are vector spaces'''. '''([[User:Leo algknt|Leo algknt]] and Jesse had a discussion.) 
'''

Let <math>\iota : \ker f \rightarrow V</math> be the inclusion of <math>\ker f</math> into <math>V</math>, then <math>V^*/\ker \iota^* \cong (\ker f)^*</math>.

We show that <math>\ker \iota^* = \mathrm{im} f^*</math>.

<center>
<math>
\begin{align} 
\ker \iota^* &= \{\phi \in V^* \;\;|\;\; \iota^*(\phi) = 0\}\\
&= \{\phi \in V^* \;\;|\;\; \phi\circ\iota = 0\}\\
&= \{\phi \in V^* \;\;|\;\; \phi|_{\ker f} = 0, \; \phi = f^*(\alpha), \; \alpha \in W^* \}\\
&= \{\phi \in V^* \;\;|\;\; \phi|_{\ker f} = 0, \; \phi = \alpha\circ f, \; \alpha \in W^* \}\\
&= \{f^*(\alpha) \in V^* \;\;|\;\; (\alpha\circ f)|_{\ker f} = 0, \; \alpha \in W^* \}\\
&= \mathrm{im} f^*.

\end{align}
</math>
</center>