Notes for AKT-140305/0:41:57 - Revision history

Leo algknt at 21:40, 30 July 2018

2018-07-30T21:40:56Z

← Older revision		Revision as of 17:40, 30 July 2018
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	'''Showing that''' <math>(\ker f)^* = V^/\mathrm{im}f^</math> '''for a linear map''' <math>f : V \rightarrow W</math>. '''([[User:Leo algknt\|Leo algknt]] and Jesse had a discussion.)		'''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.)
	'''		'''

Leo algknt at 21:37, 30 July 2018

2018-07-30T21:37:06Z

← Older revision		Revision as of 17:37, 30 July 2018
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	'''Showing that''' <math>(\ker f)^* = V^/\mathrm{im}f^</math> '''for a linear map''' <math>f : V \rightarrow W</math>. '''([[User:Leo algknt\|Leo algknt]] ~~([[User talk:Leo algknt\|talk]])~~ and Jesse had a discussion.)		'''Showing that''' <math>(\ker f)^* = V^/\mathrm{im}f^</math> '''for a linear map''' <math>f : V \rightarrow W</math>. '''([[User:Leo algknt\|Leo algknt]] and Jesse had a discussion.)
	'''		'''

Leo algknt: Created page with "'''Showing that''' $(\ker f)^* = V^/\mathrm{im}f^$ '''for a linear map''' $f : V \rightarrow W$ . '''(~~~ and Jesse had a discussion.) ''' Let $\..."$

2018-07-30T21:35:14Z

Created page with "'''Showing that''' <math>(\ker f)^* = V^*/\mathrm{im}f^*</math> '''for a linear map''' <math>f : V \rightarrow W</math>. '''(~~~ and Jesse had a discussion.) ''' Let <math>\..."

New page

'''Showing that''' <math>(\ker f)^* = V^*/\mathrm{im}f^*</math> '''for a linear map''' <math>f : V \rightarrow W</math>. '''([[User:Leo algknt|Leo algknt]] ([[User talk:Leo algknt|talk]]) 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>

Notes for AKT-140305/0:41:57 - Revision history

Leo algknt at 21:40, 30 July 2018

Leo algknt at 21:37, 30 July 2018

Leo algknt: Created page with "'''Showing that''' (\ker f)^* = V^*/\mathrm{im}f^* '''for a linear map''' f : V \rightarrow W. '''(~~~ and Jesse had a discussion.) ''' Let \..."

Leo algknt: Created page with "'''Showing that''' $(\ker f)^* = V^/\mathrm{im}f^$ '''for a linear map''' $f : V \rightarrow W$ . '''(~~~ and Jesse had a discussion.) ''' Let $\..."$