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<b>let </b><font color="Maroon" title="c1">F</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">V</font>, <font color="Olive" title="b2">W</font> being <a href="matrlin.html#V2" title="MATRLIN:attr.2">finite-dimensional</a> <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">F</font><br/> for <font color="Olive" title="b3">T</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Olive" title="b1">V</font>,<font color="Olive" title="b2">W</font>  st <font color="Olive" title="b3">T</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> holds <br/> <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Olive" title="b1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ranknull.html#K10" title="RANKNULL:func.10">rank</a> <font color="Olive" title="b3">T</font></span><br/><b>let </b><font color="Maroon" title="c2">V</font>, <font color="Maroon" title="c3">W</font> be  <a href="matrlin.html#V2" title="MATRLIN:attr.2">finite-dimensional</a> <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">F</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">T</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">W</font>  st <font color="Olive" title="b1">T</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> holds <br/> <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ranknull.html#K10" title="RANKNULL:func.10">rank</a> <font color="Olive" title="b1">T</font></span><br/><b>let </b><font color="Maroon" title="c4">T</font> be   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">W</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">T</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> implies  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ranknull.html#K10" title="RANKNULL:func.10">rank</a> <font color="Maroon" title="c4">T</font> )</span><br/>





<b>assume </b><a NAME="E1:61"/><i><font color="Green" title="E45">A1</font></i>: 
<font color="Maroon" title="c4">T</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ranknull.html#K10" title="RANKNULL:func.10">rank</a> <font color="Maroon" title="c4">T</font></span><br/>

<a NAME="E2:61"/>
 <a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <font color="Maroon" title="c4">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font>
 
<b>by </b><i><a class="txt" href="#E1:61"><i><font color="Green" title="E45">A1</font></i></a>, <a class="ref" href="ranknull.html#T15" target="_self" title="RANKNULL:th.15">Th15</a></i>;<br/>
<a NAME="E3:61"/><b>then </b><i><font color="Green" title="E46">A2</font></i>: 
 <a href="ranknull.html#K11" title="RANKNULL:func.11">nullity</a> <font color="Maroon" title="c4">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<b>by </b><i><a class="ref" href="ranknull.html#T16" target="_self" title="RANKNULL:th.16">Th16</a></i>;<br/>
 <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c2">V</font> = 
<span class="p1">(<span class="default"><a href="ranknull.html#K10" title="RANKNULL:func.10">rank</a> <font color="Maroon" title="c4">T</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="ranknull.html#K11" title="RANKNULL:func.11">nullity</a> <font color="Maroon" title="c4">T</font></span>)</span>
<b>by </b><i><a class="ref" href="ranknull.html#T44" target="_self" title="RANKNULL:th.44">Th44</a></i>
<br/>.= 
 <a href="ranknull.html#K10" title="RANKNULL:func.10">rank</a> <font color="Maroon" title="c4">T</font>
<b>by </b><i><a class="txt" href="#E3:61"><i><font color="Green" title="E46">A2</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E5:61"/>
 <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ranknull.html#K10" title="RANKNULL:func.10">rank</a> <font color="Maroon" title="c4">T</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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