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This article collects the main '''theorems and definitions in linear algebra'''.
 
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'' A '''vector space'''( or linear space) '''''V''''' over a number field² '''F''' consists of a set on which two operations (called '''addition''' and '''scalar multiplication''', respectively) are defined so, that for each pair of elements ''x'', ''y'', in '''''V''''' there is a unique element ''x'' + ''y'' in '''''V''''', and for each element a in F and each element x in '''''V''''' there is a unique element ax in '''''V''''', such that the following conditions hold.''


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*(VS 1) ''For all <math>x, y</math> in '''''V''''', <math>x+y=y+x</math>   ('''commutativity of addition''').''
 
*(VS 2) ''For all <math>x, y, z</math> in '''''V''''', <math>(x+y)+z=x+(y+z)</math>   ('''associativity of addition''').''
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*(VS 3) ''There exists an element in '''''V''''' denoted by <math>0</math> such that <math>x+0=x</math> for each <math>x</math> in '''''V'''''.''
 
*(VS 4) ''For each element <math>x</math> in '''''V''''' there exists an element <math>y</math> in '''''V''''' such that <math>x+y=0</math>.''
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*(VS 5) ''For each element <math>x</math> in '''''V''''', <math>1x=x</math>.''
 
*(VS 6) ''For each pair of element <math>a,b</math> in '''F''' and each element <math>x</math> in '''''V''''', <math>(ab)x=a(bx)</math>.
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*(VS 7) ''For each element <math>a</math> in '''F''' and each pair of elements <math>x,y</math> in '''''V''''', <math>a(x+y)=ax+ay</math>.''
 
*(VS 8) ''For each pair of elements <math>a,b</math> in '''F''' and each pair of elements <math>x</math> in '''''V''''', <math>(a+b)x=ax+bx</math>.''
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===Subspaces===
'' A '''subspace''' '''''W''''' of a '''vector space''' '''''V''''' over a field F is a '''subset''' of '''''V''''' which also has the properties that '''''W''''' is closed under scaler addition an multiplication. That is, For all ''x'', ''y'' in '''''W''''', ''x'' and ''y'' are in '''''V''''' and for any ''c'' in F, <math>cx+y</math> is in '''''W'''''.


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===Linear combinations===
 
[[Linear combination]]
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{{Empty section|date=February 2011}}


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===Linear dependence===
 
{{Empty section|date=February 2011}}
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===Linear independence===
{{Empty section|date=February 2011}}


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===Dimension===
{{Empty section|date=February 2011}}


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<!--
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===Linear transformations===
 
===Null spaces===
  <li>[http://204.13.164.121/spip.php?article4807&lang=nl/ http://204.13.164.121/spip.php?article4807&lang=nl/]</li>
===Ranges===
 
===The matrix representation of a linear transformation===
  <li>[http://enseignement-lsf.com/spip.php?article64#forum24446803 http://enseignement-lsf.com/spip.php?article64#forum24446803]</li>
===Composition of linear transformations===
 
===Matrix multiplication===
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===Invertibility===
 
===Isomorphisms===
</ul>
===The change-of-coordinates matrix===</nowiki> -->


== Nike Air Max 1 Essential  de toxicomanie ==
[[Change of coordinate matrix]]<br/>
[[Clique]]<br/>
[[Coordinate vector relative to a basis]]<br/>
[[Dimension theorem]]<br/>
[[Dominance relation]]<br/>
[[Identity matrix]]<br/>
[[Identity transformation]]<br/>
[[Incidence matrix]]<br/>
[[Inverse of a linear transformation]]<br/>
[[Inverse of a matrix]]<br/>
[[Invertible linear transformation]]<br/>
[[Isomorphic vector spaces]]<br/>
[[Isomorphism]]<br/>
[[Kronecker delta]]<br/>
[[Left-multiplication transformation]]<br/>
[[Linear operator]]<br/>
[[Linear transformation]]<br/>
[[Matrix representing a linear transformation]]<br/>
[[Nullity of a linear transformation]]<br/>
[[Null space]]<br/>
[[Ordered basis]]<br/>
[[Product of matrices]]<br/>
[[Projection on a subspace]]<br/>
[[Projection on the x-axis]]<br/>
[[Range (mathematics)|Range]]<br/>
[[Rank of a linear transformation]]<br/>
[[Reflection about the x-axis]]<br/>
[[Rotation]]<br/>
[[Similar matrices]]<br/>
Standard ordered [[basis (linear algebra)|basis]] for <math>F_n</math><br/>
[[Standard representation of a vector space with respect to a basis]]<br/>
[[Zero transformation]]


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P.S.
 
[[coefficient of the differential equation]], [[differentiability of complex function]],vector space of functions[[differential operator]], [[auxiliary polynomial]]{{disambiguation needed|date=April 2012}}, to the power of a complex number, [[exponential function]].
  <li>[http://60.191.126.74:2222/dengpin/bbs/forum.php?mod=viewthread&tid=2817498 http://60.191.126.74:2222/dengpin/bbs/forum.php?mod=viewthread&tid=2817498]</li>
 
  <li>[http://observatoiredesreligions.fr/spip.php?article7 http://observatoiredesreligions.fr/spip.php?article7]</li>
 
  <li>[http://www.thomassankara.net/spip.php?article605&date=2008-05/ http://www.thomassankara.net/spip.php?article605&date=2008-05/]</li>
 
  <li>[http://verdamilio.info/org/spip.php?article573/ http://verdamilio.info/org/spip.php?article573/]</li>
 
</ul>


== Longchamp Soldes  explained how ==
===<math>{\color{Blue}~2.1}</math> ''N''(''T'') and ''R''(''T'') are subspaces===<!-- Theorem 2.1 -->
Let ''V'' and ''W'' be vector spaces and ''I'': ''V'' → ''W'' be linear. Then ''N''(''T'') and ''R''(''T'') are subspaces of ''V'' and ''W'', respectively.


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===<math>{\color{Blue}~2.2}</math> R(T)= span of T(basis in V)===<!-- Theorem 2.2 -->
 
Let V and W be vector spaces, and let T: V→W be linear. If <math>\beta={v_1,v_2,\ldots,v_n}</math> is a basis for V, then<br/>
  <li>[http://globaldialogue2013.go.tz/gdforum/activity http://globaldialogue2013.go.tz/gdforum/activity]</li>
::<math>\mathrm{R(T)}=\mathrm{span}(T(\beta\mathrm{))}=\mathrm{span}({T(v_1),T(v_2),\ldots,T(v_n)})</math>.
 
  <li>[http://www.pierre-le-jeune.info/spip.php?article3 http://www.pierre-le-jeune.info/spip.php?article3]</li>
 
  <li>[http://www.love2shine.org.uk/activity/p/102826/ http://www.love2shine.org.uk/activity/p/102826/]</li>
 
  <li>[http://www.answers.tw/node/8#comment-3439921 http://www.answers.tw/node/8#comment-3439921]</li>
 
</ul>


== Mont Blanc Stylo Vasquez ==
===<math>{\color{Blue}~2.3}</math> Dimension theorem===<!-- Theorem 2.3 -->
Let V and W be vector spaces, and let T: V → W be linear. If V is finite-dimensional, then<br/>
::::::<math>\mathrm{nullity}(T)+\mathrm{rank}(T)=\dim(V).</math>


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===<math>{\color{Blue}~2.4}</math> one-to-one ⇔ N(T) = {0}===<!-- Theorem 2.4 -->
 
Let V and W be vector spaces, and let T: V→W be linear. Then T is one-to-one if and only if N(T)={0}.
  <li>[http://www.ovnprod.com/spip.php?article24/ http://www.ovnprod.com/spip.php?article24/]</li>
 
  <li>[http://joomacedemo.net/theme_aceshop/jf/index.php/forum/newtopic http://joomacedemo.net/theme_aceshop/jf/index.php/forum/newtopic]</li>
 
  <li>[http://mjkcos.egloos.com/3391465/ http://mjkcos.egloos.com/3391465/]</li>
 
  <li>[http://bxl.indymedia.org/spip.php?article1671&lang=nl/ http://bxl.indymedia.org/spip.php?article1671&lang=nl/]</li>
 
</ul>


== Mbt Soldes oh no ==
===<math>{\color{Blue}~2.5}</math> one-to-one ⇔ onto ⇔ rank(''T'') = dim(''V'')===<!-- Theorem 2.5 -->
Let V and W be vector spaces of equal (finite) dimension, and let ''T'':''V'' → ''W'' be linear. Then the following are equivalent.<br/>
:(a) ''T'' is one-to-one.
:(b) ''T'' is onto.
:(c) rank(''T'') = dim(''V'').


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===<math>{\color{Blue}~2.6}</math> <math>{w_1,w_2,\ldots,w_n}=</math> exactly one T (basis),===<!-- Theorem 2.6 -->
 
Let V and W be vector space over F, and suppose that <math>{v_1, v_2,\ldots,v_n}</math> is a basis for V. For <math>w_1, w_2,\ldots,w_n</math> in W, there exists exactly one linear transformation T: V→W such that <math>\mathrm{T}(v_i)=w_i</math> for <math>i=1,2,\ldots,n.</math>
  <li>[http://www.perloca.com/Main_Page http://www.perloca.com/Main_Page]</li>
<br/>
 
'''Corollary.'''
  <li>[http://verdamilio.net/tonio/spip.php?article1279/ http://verdamilio.net/tonio/spip.php?article1279/]</li>
Let V and W be vector spaces, and suppose that V has a finite basis <math>{v_1,v_2,\ldots,v_n}</math>. If U, T: V→W are linear and <math>U(v_i)=T(v_i)</math> for <math>i=1,2,\ldots,n,</math> then U=T.
 
  <li>[http://www.designer-notes.com/?p=765#comments http://www.designer-notes.com/?p=765#comments]</li>
 
  <li>[http://mjkcos.egloos.com/3391465/ http://mjkcos.egloos.com/3391465/]</li>
 
</ul>


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===<math>{\color{Blue}~2.7}</math> T is vector space===<!-- Theorem 2.7 -->
Let V and W be vector spaces over a field F, and let T, U: V→W be linear.<br/>
:(a) For all <math>a</math> &isin; ''F'', <math>a\mathrm{T}+\mathrm{U}</math> is linear.
:(b) Using the operations of addition and scalar multiplication in the preceding definition, the collection of all linear transformations form V to W is a vector space over F.


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===<math>{\color{Blue}~2.8}</math> linearity of matrix representation of linear transformation===<!-- Theorem 2.8 -->
 
Let V and W ve finite-dimensional vector spaces with ordered bases β and γ, respectively, and let T, U: V→W be linear transformations. Then<br/>
  <li>[http://219.133.37.42:8813/bbs/forum.php?mod=viewthread&tid=1873446 http://219.133.37.42:8813/bbs/forum.php?mod=viewthread&tid=1873446]</li>
:(a)<math>[T+U]_\beta^\gamma=[T]_\beta^\gamma+[U]_\beta^\gamma</math> and
 
:(b)<math>[aT]_\beta^\gamma=a[T]_\beta^\gamma</math> for all scalars <math>a</math>.
  <li>[http://www.observatoiredesreligions.fr/spip.php?article11 http://www.observatoiredesreligions.fr/spip.php?article11]</li>
 
  <li>[http://www.philatelie-france-russie.fr/spip.php?article51/ http://www.philatelie-france-russie.fr/spip.php?article51/]</li>
 
  <li>[http://www.szmoyu.com/forum.php?mod=viewthread&tid=488607 http://www.szmoyu.com/forum.php?mod=viewthread&tid=488607]</li>
 
</ul>


== Oakley France  about art and how we decide its value ==
===<math>{\color{Blue}~2.9}</math> composition law of linear operators===<!-- Theorem 2.9 -->
Let V,w, and Z be vector spaces over the same field f, and let T:V→W and U:W→Z be linear. then UT:V→Z is linear.


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===<math>{\color{Blue}~2.10}</math> law of linear operator===<!-- Theorem 2.10 -->
 
Let v be a vector space. Let T, U<sub>1</sub>, U<sub>2</sub> <math>\mathcal{L}</math>(V). Then<br/>
  <li>[http://globaldialogue2013.go.tz/gdforum/activity http://globaldialogue2013.go.tz/gdforum/activity]</li>
(a) T(U<sub>1</sub>+U<sub>2</sub>)=TU<sub>1</sub>+TU<sub>2</sub> and (U<sub>1</sub>+U<sub>2</sub>)T=U<sub>1</sub>T+U<sub>2</sub>T<br/>
 
(b) T(U<sub>1</sub>U<sub>2</sub>)=(TU<sub>1</sub>)U<sub>2</sub><br/>
  <li>[http://brussel.indymedia.org/spip.php?article3923&lang=fr/ http://brussel.indymedia.org/spip.php?article3923&lang=fr/]</li>
(c) TI=IT=T<br/>
 
(d) <math>a</math>(U<sub>1</sub>U<sub>2</sub>)=(<math>a</math>U<sub>1</sub>)U<sub>2</sub>=U<sub>1</sub>(<math>a</math>U<sub>2</sub>) for all scalars <math>a</math>.
  <li>[http://www.observatoiredesreligions.fr/spip.php?article8 http://www.observatoiredesreligions.fr/spip.php?article8]</li>
 
  <li>[http://www.butterflycluster.net/wiki/index.php?title=User:Ticgcian#Nike_Air_Max_1_Femme__apr.C3.A8s_les_appels http://www.butterflycluster.net/wiki/index.php?title=User:Ticgcian#Nike_Air_Max_1_Femme__apr.C3.A8s_les_appels]</li>
 
</ul>


== Nike Blazer  temporal statistical modeling ==
===<math>{\color{Blue}~2.11}</math> [UT]<sub>α</sub><sup>γ</sup>=[U]<sub>β</sub><sup>γ</sup>[T]<sub>α</sub><sup>β</sup>===<!-- Theorem 2.11 -->
Let V, W and Z be finite-dimensional vector spaces with ordered bases α β γ, respectively. Let T: V⇐W and U: W→Z be linear transformations. Then<br/>
:::::::<math>[UT]_\alpha^\gamma=[U]_\beta^\gamma[T]_\alpha^\beta</math>.


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'''Corollary'''. Let V be a finite-dimensional vector space with an ordered basis β. Let T,U∈<math>\mathcal{L}</math>(V). Then [UT]<sub>β</sub>=[U]<sub>β</sub>[T]<sub>β</sub>.
 
  <li>[http://passerelle.ethiopie.free.fr/spip.php?article448/ http://passerelle.ethiopie.free.fr/spip.php?article448/]</li>
 
  <li>[http://blog.myoeoe.com.tw/forum.php?mod=viewthread&tid=2563507 http://blog.myoeoe.com.tw/forum.php?mod=viewthread&tid=2563507]</li>
 
  <li>[http://verdamilio.net/tonio/spip.php?article1/ http://verdamilio.net/tonio/spip.php?article1/]</li>
 
  <li>[http://metransparent.nfrance.com/~k1001/spip.php?article8359&lang=ar&id_forum=8701/ http://metransparent.nfrance.com/~k1001/spip.php?article8359&lang=ar&id_forum=8701/]</li>
 
</ul>


== Michael Kors Sac  duration of numerous spells ==
===<math>{\color{Blue}~2.12}</math> law of matrix===<!-- Theorem 2.12 -->
Let A be an m&times;n matrix, B and C be n&times;p matrices, and D and E be q&times;m matrices. Then<br/>
:(a) A(B+C)=AB+AC and (D+E)A=DA+EA.
:(b) <math>a</math>(AB)=(<math>a</math>A)B=A(<math>a</math>B) for any scalar <math>a</math>.
:(c) I<sub>m</sub>A=AI<sub>m</sub>.
:(d) If V is an n-dimensional vector space with an ordered basis &beta;, then [I<sub>v</sub>]<sub>&beta;</sub>=I<sub>n</sub>.


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'''Corollary.''' Let A be an m&times;n matrix, B<sub>1</sub>,B<sub>2</sub>,...,B<sub>k</sub> be n&times;p matrices, C<sub>1</sub>,C<sub>1</sub>,...,C<sub>1</sub> be q&times;m matrices, and <math>a_1,a_2,\ldots,a_k</math> be scalars. Then<br/>
 
:::::::<math>A\Bigg(\sum_{i=1}^k a_iB_i\Bigg)=\sum_{i=1}^k a_iAB_i</math>
  <li>[http://bbsznds.0371-net.com/forum.php?mod=viewthread&tid=291909 http://bbsznds.0371-net.com/forum.php?mod=viewthread&tid=291909]</li>
and
 
:::::::<math>\Bigg(\sum_{i=1}^k a_iC_i\Bigg)A=\sum_{i=1}^k a_iC_iA</math>.
  <li>[http://tztea.com.cn/news/html/?438440.html http://tztea.com.cn/news/html/?438440.html]</li>
 
  <li>[http://moban.l5w.cn/news/html/?1156854.html http://moban.l5w.cn/news/html/?1156854.html]</li>
 
  <li>[http://ciarcr.org/spip.php?article310/ http://ciarcr.org/spip.php?article310/]</li>
 
</ul>


== New Balance Homme  leading.. ==
===<math>{\color{Blue}~2.13}</math> law of column multiplication===<!-- Theorem 2.13 -->
Let A be an m&times;n matrix and B be an n&times;p matrix. For each <math>j (1\le j\le p)</math> let <math>u_j</math> and <math>v_j</math> denote the jth columns of AB and B, respectively. Then<br/>
(a) <math>u_j=Av_j</math><br/>
(b) <math>v_j=Be_j</math>, where <math>e_j</math> is the jth standard vector of F<sup>p</sup>.


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===<math>{\color{Blue}~2.14}</math> [T(u)]<sub>γ</sub>=[T]<sub>β</sub><sup>γ</sup>[u]<sub>β</sub>===<!-- Theorem 2.14 -->
 
Let V and W be finite-dimensional vector spaces having ordered bases β and γ, respectively, and let T: V→W be linear. Then, for each u ∈ V, we have<br/>
  <li>[http://enseignement-lsf.com/spip.php?article64#forum25127235 http://enseignement-lsf.com/spip.php?article64#forum25127235]</li>
::::::::<math>[T(u)]_\gamma=[T]_\beta^\gamma[u]_\beta</math>.
 
  <li>[http://metransparent.nfrance.com/~k1001/spip.php?article8359&lang=ar&id_forum=8701/ http://metransparent.nfrance.com/~k1001/spip.php?article8359&lang=ar&id_forum=8701/]</li>
 
  <li>[http://cgi.members.interq.or.jp/silver/ginkuji/skin/001/apeboard.cgi?command=read_message/ http://cgi.members.interq.or.jp/silver/ginkuji/skin/001/apeboard.cgi?command=read_message/]</li>
 
  <li>[http://www.wester.net.cn/bbs/forum.php?mod=viewthread&tid=64598 http://www.wester.net.cn/bbs/forum.php?mod=viewthread&tid=64598]</li>
 
</ul>


== Louis Vuitton Sneakers In instances of dementia for example ==
===<math>{\color{Blue}~2.15}</math> laws of L<sub>A</sub>===<!-- Theorem 2.15 -->
Let A be an m×n matrix with entries from F. Then the left-multiplication transformation L<sub>A</sub>: F<sup>n</sup>→F<sup>m</sup> is linear. Furthermore, if B is any other m&times;n matrix (with entries from F) and β and γ are the standard ordered bases for F<sup>n</sup> and F<sup>m</sup>, respectively, then we have the following properties.<br/>
(a) <math>[L_A]_\beta^\gamma=A</math>.<br/>
(b) L<sub>A</sub>=L<sub>B</sub> if and only if A=B.<br/>
(c) L<sub>A+B</sub>=L<sub>A</sub>+L<sub>B</sub> and L<sub><math>a</math>A</sub>=<math>a</math>L<sub>A</sub> for all <math>a</math>∈F.<br/>
(d) If T:F<sup>n</sup>→F<sup>m</sup> is linear, then there exists  a unique m&times;n matrix C such that T=L<sub>C</sub>. In fact, <math>\mathrm{C}=[L_A]_\beta^\gamma</math>.<br/>
(e) If W is an n&times;p matrix, then L<sub>AE</sub>=L<sub>A</sub>L<sub>E</sub>.<br/>
(f ) If m=n, then <math>L_{I_n}=I_{F^n}</math>.


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===<math>{\color{Blue}~2.16}</math> A(BC)=(AB)C===<!-- Theorem 5.16 -->
 
Let A,B, and C be matrices such that A(BC) is defined. Then A(BC)=(AB)C; that is, matrix multiplication is associative.
  <li>[http://www.dpcameron.co.uk/index.php?option=com_kunena&func=view&catid=3&id=119243&Itemid=0#119243 http://www.dpcameron.co.uk/index.php?option=com_kunena&func=view&catid=3&id=119243&Itemid=0#119243]</li>
 
  <li>[http://observatoiredesreligions.fr/spip.php?article9 http://observatoiredesreligions.fr/spip.php?article9]</li>
 
  <li>[http://cerisier.info/spip.php?article25/ http://cerisier.info/spip.php?article25/]</li>
 
  <li>[http://www.observatoiredesreligions.fr/spip.php?article8 http://www.observatoiredesreligions.fr/spip.php?article8]</li>
 
</ul>


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===<math>{\color{Blue}~2.17}</math> T<sup>-1</sup>is linear===
Let V and W be vector spaces, and let T:V→W be linear and invertible. Then T<sup>−1</sup>: W
→V is linear.


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===<math>{\color{Blue}~2.18}</math> [T<sup>-1</sup>]<sub>&gamma;</sub><sup>&beta;</sup>=([T]<sub>&beta;</sub><sup>&gamma;</sup>)<sup>-1</sup>===
 
Let V and W be finite-dimensional vector spaces with ordered bases β and γ, respectively. Let T:V→W be linear. Then T is invertible if and only if <math>[T]_\beta^\gamma</math> is invertible. Furthermore, <math>[T^{-1}]_\gamma^\beta=([T]_\beta^\gamma)^{-1}</math>
  <li>[http://www.film-video-dvd-production.com/spip.php?article6/ http://www.film-video-dvd-production.com/spip.php?article6/]</li>
 
 
'''Lemma.''' Let T be an invertible linear transformation from V to W. Then V is finite-dimensional if and only if W is finite-dimensional. In this case, dim(V)=dim(W).
  <li>[http://www.philatelie-france-russie.fr/spip.php?article51/ http://www.philatelie-france-russie.fr/spip.php?article51/]</li>
 
 
'''Corollary 1.''' Let V be a finite-dimensional vector space with an ordered basis β, and let T:V→V be linear. Then T is invertible if and only if [T]<sub>β</sub> is invertible. Furthermore, [T<sup>−1</sup>]<sub>β</sub>=([T]<sub>β</sub>)<sup>−1</sup>.
  <li>[http://thomassankara.net/spip.php?article878&lang=itcon&date=2015-07&tri=date_time/ http://thomassankara.net/spip.php?article878&lang=itcon&date=2015-07&tri=date_time/]</li>
 
 
'''Corollary 2.''' Let A be an n&times;n matrix. Then A is invertible if and only if L<sub>A</sub> is invertible. Furthermore, (L<sub>A</sub>)<sup>−1</sup>=L<sub>A<sup>−1</sup></sub>.
  <li>[http://passerelle.ethiopie.free.fr/spip.php?article81/ http://passerelle.ethiopie.free.fr/spip.php?article81/]</li>
 
 
===<math>{\color{Blue}~2.19}</math> V is isomorphic to W &hArr; dim(V)=dim(W)===
</ul>
Let W and W be finite-dimensional vector spaces (over the same field). Then V is isomorphic to W if and only if dim(V)=dim(W).
 
'''Corollary.''' Let V be a vector space over F. Then V is isomorphic to F<sup>n</sup> if and only if dim(V)=n.
 
===<math>{\color{Blue}~2.20}</math> ??===
Let  V and W be finite-dimensional vector spaces over F of dimensions n and m, respectively, and let β and γ be ordered bases for V and W, respectively. Then the function <math>~\Phi</math>: <math>\mathcal{L}</math>(V,W)→M<sub>m&times;n</sub>(F), defined by <math>~\Phi(T)=[T]_\beta^\gamma</math> for T∈<math>\mathcal{L}</math>(V,W), is an isomorphism.
 
'''Corollary.''' Let V and W be finite-dimensional vector spaces of dimension n and m, respectively. Then <math>\mathcal{L}</math>(V,W) is finite-dimensional of dimension mn.
 
===<math>{\color{Blue}~2.21}</math> ''&Phi;<sub>&beta;</sub>'' is an isomorphism===
For any finite-dimensional vector space V with ordered basis β, ''Φ<sub>β</sub>'' is an isomorphism.
 
===<math>{\color{Blue}~2.22}</math> ??===
Let β and β' be two ordered bases for a finite-dimensional vector space V, and let <math>Q=[I_V]_{\beta'}^\beta</math>. Then<br/>
(a) <math>Q</math> is invertible.<br/>
(b) For any <math>v\in</math> V, <math>~[v]_\beta=Q[v]_{\beta'}</math>.
 
===<math>{\color{Blue}~2.23}</math> [T]<sub>&beta;'</sub>=Q<sup>-1</sup>[T]<sub>&beta;</sub>Q===
Let T be a linear operator on a finite-dimensional vector space V,and let β and β' be two ordered bases for V. Suppose that Q is the change of coordinate matrix that changes β'-coordinates into β-coordinates. Then<br/>
:::::::<math>~[T]_{\beta'}=Q^{-1}[T]_\beta Q</math>.
 
'''Corollary.''' Let A∈M<sub>n&times;n</sub>(''F''), and le t γ be an ordered basis for F<sup>n</sup>. Then [L<sub>A</sub>]<sub>γ</sub>=Q<sup>−1</sup>AQ, where Q is the n&times;n matrix whose jth column is the jth vector of γ.
 
===<math>{\color{Blue}~2.24}</math>===
{{Empty section|date=February 2011}}
 
===<math>{\color{Blue}~2.25}</math>===
{{Empty section|date=February 2011}}
 
===<math>{\color{Blue}~2.26}</math>===
{{Empty section|date=February 2011}}
 
===<math>{\color{Blue}~2.27}</math> ''p''(D)(x)=0 (''p''(D)∈C<sup>∞</sup>)⇒ x<sup>(k)</sup>exists (k∈N)===<!-- Theorem 2.27 -->
Any solution to a homogeneous linear differential equation with constant coefficients has derivatives of all orders; that is, if <math>x</math> is a solution to  such an equation, then <math>x^{(k)}</math> exists for every positive integer k.
 
===<math>{\color{Blue}~2.28}</math> {solutions}= N(p(D))===<!-- Theorem 2.28 -->
The set of all solutions to a homogeneous linear differential equation with constant coefficients coincides with the null space of p(D), where p(t) is the auxiliary polynomial with the equation.
 
'''Corollary'''. The set of all solutions to s homogeneous linear differential equation with constant coefficients is a subspace of <math>\mathrm{C}^\infty</math>.
 
===<math>{\color{Blue}~2.29}</math> derivative of exponential function===<!-- Theorem 2.29 -->
For any exponential function <math>f(t)=e^{ct}, f'(t)=ce^{ct}</math>.
 
===<math>{\color{Blue}~2.30}</math> {e<sup>-at</sup>} is a basis of N(''p''(D+aI))===<!-- Theorem 2.30 -->
The solution space for the differential equation,<br/>
::::<math>y'+a_0y=0</math>
is of dimension 1 and has <math>\{e^{-a_0t}\}</math>as a basis.
 
'''Corollary.''' For any complex number c, the null space of the differential operator D-cI has {<math>e^{ct}</math>} as a basis.
 
===<math>{\color{Blue}~2.31}</math> <math>e^{ct}</math> is a solution ===<!-- Theorem 2.31 -->
Let p(t) be the auxiliary polynomial for a homogeneous linear differential equation with constant coefficients. For any complex number c, if c is a zero of p(t), then to the differential equation.
 
===<math>{\color{Blue}~2.32}</math> dim(N(''p''(D)))=n===<!-- Theorem 2.32-->
For any differential operator p(D) of order n, the null space of p(D) is an n_dimensional subspace of C<sup>∞</sup>.
 
'''Lemma 1'''. The differential operator D-cI: C<sup>∞</sup> to C<sup></sup> is onto for any complex number c.
 
'''Lemma 2''' Let V be a vector space, and suppose that T and U are linear operators on V such that U is onto and the null spaces of T and U are finite-dimensional, Then the null space of TU is finite-dimensional, and<br/>
:::::dim(N(TU))=dim(N(U))+dim(N(U)).
 
'''Corollary'''. The solution space of any nth-order homogeneous linear differential equation with constant coefficients is an n-dimensional subspace of C<sup>∞</sup>.
 
===<math>{\color{Blue}~2.33}</math> e<sup>c<sub>i</sub>t</sup> is linearly independent with each other (c<sub>i</sub> are distinct)===<!-- Theorem 2.33 -->
Given n distinct complex numbers <math>c_1, c_2,\ldots,c_n</math>, the set of exponential functions <math>\{e^{c_1t},e^{c_2t},\ldots,e^{c_nt}\}</math> is linearly independent.
 
'''Corollary'''. For any nth-order homogeneous linear differential equation with constant coefficients, if the auxiliary polynomial has n distinct zeros <math>c_1, c_2, \ldots, c_n</math>, then <math>\{e^{c_1t},e^{c_2t},\ldots,e^{c_nt}\}</math> is a basis for the solution space of the differential equation.
 
'''Lemma'''. For a given complex number c and positive integer n, suppose that (t-c)^n is athe auxiliary polynomial of a homogeneous linear differential  equation with constant coefficients. Then the set<br/>
:::<math>\beta=\{e^{c_1t},e^{c_2t},\ldots,e^{c_nt}\}</math>
is a basis for the solution space of the equation.
 
===<math>{\color{Blue}~2.34}</math> general solution of homogeneous linear differential equation===<!-- Theorem 2.34 -->
Given a homogeneous linear differential equation with constant coefficients and auxiliary polynomial<br/>
:::::<math>(t-c_1)^n_1(t-c_2)^n_2\cdots(t-c_k)^n_k,</math>
where <math>n_1, n_2,\ldots,n_k</math> are positive integers and <math>c_1, c_2, \ldots, c_n</math> are distinct complex numbers, the following set is a basis for the solution space of the equation:<br/>
:::<math>\{e^{c_1t}, te^{c_1t},\ldots,t^{n_1-1}e^{c_1t},\ldots,e{c_kt},te^{c_kt},\ldots,t^{n_k-1}e^{c_kt}\}</math>.
 
==Elementary matrix operations and systems of linear equations==
===Elementary matrix operations===
{{Empty section|date=February 2011}}
 
===Elementary matrix===
{{Empty section|date=February 2011}}
 
===Rank of a matrix===
The rank of a matrix A is the dimension of the column space of A.
 
===Matrix inverses===
{{Empty section|date=February 2011}}
 
===System of linear equations===
{{Empty section|date=February 2011}}
 
==Determinants==
 
''If''
::::::::: <math>A = \begin{pmatrix}
a & b \\
c & d \\
\end{pmatrix}</math>
''is a'' 2×2'' matrix with entries form a field F, then we define the '''determinant''' of A, denoted ''det(''A'')'' or |A|, to be the scalar <math>ad-bc</math>.''
 
<!-- P173 of Stephen -->
 
*Theorem 1: linear function for a single row.<br />
*Theorem 2: nonzero determinant ⇔ invertible matrix
 
'''Theorem 1:'''
'' The function ''det: '''M'''<sub>2×2</sub>(''F'')'' → F is a linear function of each row of a ''2×2'' matrix when the other row is held fixed. That is, if <math>u,v,</math> and <math>w</math> are in '''''F'''²'' and <math>k</math> is a scalar, then''
:::::::<math>\det\begin{pmatrix}
u + kv\\
w\\
\end{pmatrix}
=\det\begin{pmatrix}
u\\
w\\
\end{pmatrix}
+ k\det\begin{pmatrix}
v\\
w\\
\end{pmatrix}</math>
 
''and''
 
:::::::<math>\det\begin{pmatrix}
w\\
u + kv\\
\end{pmatrix}
=\det\begin{pmatrix}
w\\
u\\
\end{pmatrix}
+ k\det\begin{pmatrix}
w\\
v\\
\end{pmatrix}</math>
 
'''Theorem 2''':
''Let A <math>\in</math> ''M<sub>2×2</sub>(''F'')''. Then thee deter minant of A is nonzero if and only if A is invertible. Moreover, if A is invertible, then''<br/>
::::::::<math>A^{-1}=\frac{1}{\det(A)}\begin{pmatrix}
A_{22}&-A_{12}\\
-A_{21}&A_{11}\\
\end{pmatrix}</math>
 
==Diagonalization==
[[Characteristic polynomial]] of a linear operator/matrix
 
===<math>{\color{Blue}~5.1}</math> diagonalizable⇔basis of eigenvector===<!-- Theorem 5.1 -->
A linear operator T on a finite-dimensional vector space V is diagonalizable if and only if there exists an ordered basis β for V consisting of eigenvectors of T. Furthermore, if T is diagonalizable, <math>\beta= {v_1,v_2,\ldots,v_n}</math> is an ordered basis of eigenvectors of T, and ''D'' = [T]<sub>β</sub> then D is a diagonal matrix and <math>D_{jj}</math> is the eigenvalue corresponding to <math>v_j</math> for <math>1\le j \le n</math>.
 
===<math>{\color{Blue}~5.2}</math> eigenvalue⇔det(''A''-λ''I''n)=0===<!-- Theorem 5.2 -->
Let ''A''∈M<sub>n×n</sub>(''F''). Then a scalar λ is an eigenvalue of ''A'' if and only if det(''A''-λ''I''<sub>n</sub>)=0
 
===<math>{\color{Blue}~5.3}</math> characteristic polynomial===<!-- Theorem 5.3 -->
Let A∈Mn&times;n(''F'').<br/>
(a) The characteristic polynomial of A is a polynomial of degree n with leading coefficient(-1)n.<br/>
(b) A has at most n distinct eigenvalues.
 
===<math>{\color{Blue}~5.4}</math> υ to λ⇔υ∈N(T-λI)===<!-- Theorem 5.4 -->
Let T be a linear operator on a vector space V, and let λ be an eigenvalue of T. <br/>A vector υ∈V is an eigenvector of T corresponding to λ if and only if υ≠0 and υ∈N(T-λI).
 
===<math>{\color{Blue}~5.5}</math> vi to λi⇔vi is linearly independent===<!-- Theorem 5.5 -->
Let T be a linear operator on a vector space V, and let <math>\lambda_1,\lambda_2,\ldots,\lambda_k,</math> be distinct eigenvalues of T. If <math>v_1,v_2,\ldots,v_k</math> are eigenvectors of t such that <math>\lambda_i</math> corresponds to <math>v_i</math> (<math>1\le i\le k</math>), then {<math>v_1,v_2,\ldots,v_k</math>} is linearly independent.
 
===<math>{\color{Blue}~5.6}</math> characteristic polynomial splits===<!-- Theorem 5.6 -->
The characteristic polynomial of any diagonalizable linear operator splits.
 
===<math>{\color{Blue}~5.7}</math> 1 ≤ dim(E''λ'') ≤ ''m''===<!-- Theorem 5.7 -->
Let T be alinear operator on a finite-dimensional vectorspace V, and let λ be an eigenvalue of T having multiplicity <math>m</math>. Then <math>1 \le\dim(E_{\lambda})\le m</math>.
 
===<math>{\color{Blue}~5.8}</math> ''S'' = S<sub>1</sub> ∪ ''S''<sub>2</sub> ∪ ...∪ ''S''<sub>''k''</sub> is linearly independent===<!-- Theorem 5.8 -->
Let T be a linear operator on a vector space V, and let <math>\lambda_1,\lambda_2,\ldots,\lambda_k,</math> be distinct eigenvalues of T. For each <math>i=1,2,\ldots,k,</math> let <math>S_i</math> be a finite linearly independent subset of the eigenspace <math>E_{\lambda_i}</math>. Then <math>S=S_1\cup S_2 \cup\cdots\cup S_k</math> is a linearly independent subset of V.
 
===<math>{\color{Blue}~5.9}</math> ⇔T is diagonalizable===<!-- Theorem 5.9 -->
Let T be a linear operator on a finite-dimensional vector space V that the characteristic polynomial of T splits. Let <math>\lambda_1,\lambda_2,\ldots,\lambda_k</math> be the distinct eigenvalues of T. Then<br/>
(a) T is diagonalizable if and only if the multiplicity of <math>\lambda_i</math> is equal to <math>\dim(E_{\lambda_i})</math> for all <math>i</math>.<br/>
(b) If T is diagonalizable and <math>\beta_i</math> is an ordered basis for <math>E_{\lambda_i}</math> for each <math>i</math>, then <math>\beta=\beta_1\cup \beta_2\cup \cup\beta_k</math> is an ordered <math>basis^2</math> for V consisting of eigenvectors of T.
 
'''Test for diagonlization'''
 
==Inner product spaces==
[[Inner product]], [[standard inner product]] on F<sup>n</sup>, [[conjugate transpose]], [[adjoint]]{{dn|date=December 2013}}, [[Frobenius inner product]], complex/real [[inner product space]], [[Norm (mathematics)|norm]], [[length]], [[conjugate linear]], [[orthogonal]], [[perpendicular]], [[orthogonal]], [[unit vector]], [[orthonormal]], [[normalized vector|normalization]].
 
===<math>{\color{Blue}~6.1}</math> properties of linear product===
Let V be an inner product space. Then for x,y,z\in V and c \in f, the following staements are true.<br/>
(a) <math>\langle x,y+z\rangle=\langle x,y\rangle+\langle x,z\rangle.</math><br/>
(b) <math>\langle x,cy\rangle=\bar{c}\langle x,y\rangle.</math><br/>
(c) <math>\langle x,\mathit{0}\rangle=\langle\mathit{0},x\rangle=0.</math><br/>
(d) <math>\langle x,x\rangle=0</math> if and only if <math>x=\mathit{0}.</math><br/>
(e) If<math>\langle x,y\rangle=\langle x,z\rangle</math> for all <math>x\in</math> V, then <math>y=z</math>.
 
===<math>{\color{Blue}~6.2}</math> law of norm===
Let V be an inner product space over F. Then for all x,y\in V and c\in F, the following statements are true.<br/>
(a) <math>\|cx\|=|c|\cdot\|x\|</math>.<br/>
(b) <math>\|x\|=0 </math> if and only if <math>x=0</math>. In any case, <math>\|x\|\ge0</math>.<br/>
(c)('''Cauchy-Schwarz In equality''')<math>|\langle x,y\rangle|\le\|x\|\cdot\|y\|</math>.<br/>
(d)('''Triangle Inequality''')<math>\|x+y\|\le\|x\|+\|y\|</math>.
 
[[orthonormal basis]], [[Gram–Schmidt process]], [[Fourier coefficients]], [[orthogonal complement]], [[orthogonal projection]]
 
===<math>{\color{Blue}~6.3}</math> span of orthogonal subset===
Let V be an inner product space and <math>S=\{v_1,v_2,\ldots,v_k\}</math> be an orthogonal subset of V consisting of nonzero vectors. If <math>y</math>∈span(S), then<br/>
::::::<math>y=\sum_{i=1}^n{\langle y,v_i \rangle \over \|v_i\|^2}v_i</math>
 
===<math>{\color{Blue}~6.4}</math> Gram-Schmidt process===
Let V be an inner product space and S=<math>\{w_1,w_2,\ldots,w_n\}</math> be a linearly independent subset of V. DefineS'=<math>\{v_1,v_2,\ldots,v_n\}</math>, where <math>v_1=w_1</math> and<br/>
::::::<math>v_k=w_k-\sum_{j=1}^{k-1}{\langle w_k, v_j\rangle\over\|v_j\|^2}v_j</math>
Then S' is an orhtogonal set of nonzero vectors such that span(S')=span(S).
 
===<math>{\color{Blue}~6.5}</math> orthonormal basis===
Let V be a nonzero finite-dimensional inner product space. Then V has an orthonormal basis β. Furthermore, if β =<math>\{v_1,v_2,\ldots,v_n\}</math> and x∈V, then <br/>
::::::<math>x=\sum_{i=1}^n\langle x,v_i\rangle v_i</math>.
 
'''Corollary.''' Let V be a finite-dimensional inner product space with an orthonormal basis β =<math>\{v_1,v_2,\ldots,v_n\}</math>. Let T be a linear operator on V, and let A=[T]<sub>β</sub>. Then for any <math>i</math> and <math>j</math>, <math>A_{ij}=\langle T(v_j), v_i\rangle</math>.
 
===<math>{\color{Blue}~6.6}</math> W<sup>&perp;</sup> by orthonormal basis===
Let W be a finite-dimensional subspace of an inner product space V, and let <math>y</math>∈V. Then there exist unique vectors <math>u</math>∈W and <math>z</math>∈W<sup>⊥</sup> such that <math>y=u+z</math>. Furthermore, if <math>\{v_1,v_2,\ldots,v_k\}</math> is an orthornormal basis for W, then <br/>
::::::<math>u=\sum_{i=1}^k\langle y,v_i\rangle v_i</math>.
S=\{v_1,v_2,\ldots,v_k\}
'''Corollary.''' In the notation of Theorem 6.6, the vector <math>u</math> is the unique vector in W that is "closest" to <math>y</math>; thet is, for any <math>x</math>∈W, <math>\|y-x\|\ge\|y-u\|</math>, and this inequality is an equality if and onlly if <math>x=u</math>.
 
===<math>{\color{Blue}~6.7}</math> properties of orthonormal set===
Suppose that <math>S=\{v_1,v_2,\ldots,v_k\}</math> is an orthonormal set in an <math>n</math>-dimensional inner product space V. Than <br/>
(a) S can be extended to an orthonormal basis <math>\{v_1, v_2, \ldots,v_k,v_{k+1},\ldots,v_n\}</math> for V.<br/>
(b) If W=span(S), then <math>S_1=\{v_{k+1},v_{k+2},\ldots,v_n\}</math> is an orhtonormal basis for W<sup>⊥</sup>(using the preceding notation).<br/>
(c) If W is any subspace of V, then dim(V)=dim(W)+dim(W<sup>⊥</sup>).
 
[[Least squares approximation]], [[Minimal solutions to systems of linear equations]]
 
===<math>{\color{Blue}~6.8}</math> linear functional representation inner product===
Let V be a finite-dimensional inner product space over F, and let <math>g</math>:V→F be a linear transformation. Then there exists a unique vector <math>y</math>∈ V such that <math>\rm{g}(x)=\langle x, y\rangle</math> for all <math>x</math>∈ V.
 
===<math>{\color{Blue}~6.9}</math> definition of T*===
Let V be a finite-dimensional inner product space, and let T be a linear operator on V. Then there exists a unique function T*:V→V such that <math>\langle\rm{T}(x),y\rangle=\langle x, \rm{T}^*(y)\rangle</math> for all <math>x,y</math> ∈ V. Furthermore, T* is linear
 
===<math>{\color{Blue}~6.10}</math> [T*]<sub>&beta;</sub>=[T]*<sub>&beta;</sub>===
Let V be a finite-dimensional inner product space, and let β be an orthonormal basis for V. If T is a linear operator on V, then<br/>
:::::<math>[T^*]_\beta=[T]^*_\beta</math>.
 
===<math>{\color{Blue}~6.11}</math> properties of T*===
Let V be an inner product space, and let T and U be linear operators onV. Then<br/>
(a) (T+U)*=T*+U*;<br/>
(b) (<math>c</math>T)*=<math>\bar c</math> T* for any c∈ F;<br/>
(c) (TU)*=U*T*;<br/>
(d) T**=T;<br/>
(e) I*=I.
 
'''Corollary.''' Let A and B be n&times;nmatrices. Then <br/>
(a) (''A''+''B'')*=''A''*+''B''*;<br/>
(b) (<math>c</math>''A'')*=<math>\bar c</math> ''A''* for any <math>c</math>∈ F;<br/>
(c) (''AB'')*=''B''*''A''*;<br/>
(d) ''A''**=''A'';<br/>
(e) ''I''*=''I''.
 
===<math>{\color{Blue}~6.12}</math> Least squares approximation===
Let ''A'' ∈ M<sub>m&times;n</sub>(''F'') and <math>y</math>∈F<sup>m</sup>. Then there exists <math>x_0</math> ∈ F<sup>n</sup> such that <math>(A*A)x_0=A*y</math> and <math>\|Ax_0-Y\|\le\|Ax-y\|</math> for all x∈ F<sup>n
</sup>
 
'''Lemma 1.''' let ''A ''∈ M<sub>m&times;n</sub>(''F''), <math>x</math>∈F<sup>n</sup>, and <math>y</math>∈F<sup>m</sup>. Then <br/>
:::::<math>\langle Ax, y\rangle _m =\langle x, A*y\rangle _n</math>
 
'''Lemma 2.''' Let ''A ''∈ M<sub>m&times;n</sub>(''F''). Then rank(''A*A'')=rank(''A'').
 
'''Corollary.'''(of lemma 2) If ''A'' is an m&times;n matrix such that rank(''A'')=n, then ''A*A'' is invertible.
 
===<math>{\color{Blue}~6.13}</math> Minimal solutions to systems of linear equations===
Let ''A ''∈ M<sub>m&times;n</sub>(''F'') and b∈ F<sup>m</sup>. Suppose that <math>Ax=b</math> is consistent. Then the following statements are true.<br/>
(a) There existes exactly one minimal solution <math>s</math> of <math>Ax=b</math>, and <math>s</math>∈R(L<sub>''A''*</sub>).<br/>
(b) The vector <math>s</math> is the only solution to <math>Ax=b</math> that lies in R(L<sub>''A''*</sub>); that is, if <math>u</math> satisfies <math>(AA*)u=b</math>, then <math>s=A*u</math>.
 
==Canonical forms==
{{Empty section|date=July 2010}}
 
==References==
* Linear Algebra 4th edition, by Stephen H. Friedberg Arnold J. Insel and Lawrence E. spence  ISBN 7-04-016733-6
* Linear Algebra 3rd edition, by Serge Lang (UTM)  ISBN 0-387-96412-6
* Linear Algebra and Its Applications 4th edition, by Gilbert Strang  ISBN 0-03-010567-6
 
{{DEFAULTSORT:Theorems And Definitions In Linear Algebra}}
[[Category:Linear algebra| ]]
[[Category:Theorems in linear algebra| ]]

Latest revision as of 00:18, 18 July 2013

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This article collects the main theorems and definitions in linear algebra.

Vector spaces

A vector space( or linear space) V over a number field² F consists of a set on which two operations (called addition and scalar multiplication, respectively) are defined so, that for each pair of elements x, y, in V there is a unique element x + y in V, and for each element a in F and each element x in V there is a unique element ax in V, such that the following conditions hold.

Subspaces

A subspace W of a vector space V over a field F is a subset of V which also has the properties that W is closed under scaler addition an multiplication. That is, For all x, y in W, x and y are in V and for any c in F, is in W.

Linear combinations

Linear combination

Systems of linear equations

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Linear dependence

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Linear independence

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Bases

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Dimension

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Linear transformations and matrices

Change of coordinate matrix
Clique
Coordinate vector relative to a basis
Dimension theorem
Dominance relation
Identity matrix
Identity transformation
Incidence matrix
Inverse of a linear transformation
Inverse of a matrix
Invertible linear transformation
Isomorphic vector spaces
Isomorphism
Kronecker delta
Left-multiplication transformation
Linear operator
Linear transformation
Matrix representing a linear transformation
Nullity of a linear transformation
Null space
Ordered basis
Product of matrices
Projection on a subspace
Projection on the x-axis
Range
Rank of a linear transformation
Reflection about the x-axis
Rotation
Similar matrices
Standard ordered basis for
Standard representation of a vector space with respect to a basis
Zero transformation

P.S. coefficient of the differential equation, differentiability of complex function,vector space of functionsdifferential operator, auxiliary polynomialTemplate:Disambiguation needed, to the power of a complex number, exponential function.

N(T) and R(T) are subspaces

Let V and W be vector spaces and I: VW be linear. Then N(T) and R(T) are subspaces of V and W, respectively.

R(T)= span of T(basis in V)

Let V and W be vector spaces, and let T: V→W be linear. If is a basis for V, then

.

Dimension theorem

Let V and W be vector spaces, and let T: V → W be linear. If V is finite-dimensional, then

one-to-one ⇔ N(T) = {0}

Let V and W be vector spaces, and let T: V→W be linear. Then T is one-to-one if and only if N(T)={0}.

one-to-one ⇔ onto ⇔ rank(T) = dim(V)

Let V and W be vector spaces of equal (finite) dimension, and let T:VW be linear. Then the following are equivalent.

(a) T is one-to-one.
(b) T is onto.
(c) rank(T) = dim(V).

exactly one T (basis),

Let V and W be vector space over F, and suppose that is a basis for V. For in W, there exists exactly one linear transformation T: V→W such that for
Corollary. Let V and W be vector spaces, and suppose that V has a finite basis . If U, T: V→W are linear and for then U=T.

T is vector space

Let V and W be vector spaces over a field F, and let T, U: V→W be linear.

(a) For all F, is linear.
(b) Using the operations of addition and scalar multiplication in the preceding definition, the collection of all linear transformations form V to W is a vector space over F.

linearity of matrix representation of linear transformation

Let V and W ve finite-dimensional vector spaces with ordered bases β and γ, respectively, and let T, U: V→W be linear transformations. Then

(a) and
(b) for all scalars .

composition law of linear operators

Let V,w, and Z be vector spaces over the same field f, and let T:V→W and U:W→Z be linear. then UT:V→Z is linear.

law of linear operator

Let v be a vector space. Let T, U1, U2(V). Then
(a) T(U1+U2)=TU1+TU2 and (U1+U2)T=U1T+U2T
(b) T(U1U2)=(TU1)U2
(c) TI=IT=T
(d) (U1U2)=(U1)U2=U1(U2) for all scalars .

[UT]αγ=[U]βγ[T]αβ

Let V, W and Z be finite-dimensional vector spaces with ordered bases α β γ, respectively. Let T: V⇐W and U: W→Z be linear transformations. Then

.

Corollary. Let V be a finite-dimensional vector space with an ordered basis β. Let T,U∈(V). Then [UT]β=[U]β[T]β.

law of matrix

Let A be an m×n matrix, B and C be n×p matrices, and D and E be q×m matrices. Then

(a) A(B+C)=AB+AC and (D+E)A=DA+EA.
(b) (AB)=(A)B=A(B) for any scalar .
(c) ImA=AIm.
(d) If V is an n-dimensional vector space with an ordered basis β, then [Iv]β=In.

Corollary. Let A be an m×n matrix, B1,B2,...,Bk be n×p matrices, C1,C1,...,C1 be q×m matrices, and be scalars. Then

and

.

law of column multiplication

Let A be an m×n matrix and B be an n×p matrix. For each let and denote the jth columns of AB and B, respectively. Then
(a)
(b) , where is the jth standard vector of Fp.

[T(u)]γ=[T]βγ[u]β

Let V and W be finite-dimensional vector spaces having ordered bases β and γ, respectively, and let T: V→W be linear. Then, for each u ∈ V, we have

.

laws of LA

Let A be an m×n matrix with entries from F. Then the left-multiplication transformation LA: Fn→Fm is linear. Furthermore, if B is any other m×n matrix (with entries from F) and β and γ are the standard ordered bases for Fn and Fm, respectively, then we have the following properties.
(a) .
(b) LA=LB if and only if A=B.
(c) LA+B=LA+LB and LA=LA for all ∈F.
(d) If T:Fn→Fm is linear, then there exists a unique m×n matrix C such that T=LC. In fact, .
(e) If W is an n×p matrix, then LAE=LALE.
(f ) If m=n, then .

A(BC)=(AB)C

Let A,B, and C be matrices such that A(BC) is defined. Then A(BC)=(AB)C; that is, matrix multiplication is associative.

T-1is linear

Let V and W be vector spaces, and let T:V→W be linear and invertible. Then T−1: W →V is linear.

[T-1]γβ=([T]βγ)-1

Let V and W be finite-dimensional vector spaces with ordered bases β and γ, respectively. Let T:V→W be linear. Then T is invertible if and only if is invertible. Furthermore,

Lemma. Let T be an invertible linear transformation from V to W. Then V is finite-dimensional if and only if W is finite-dimensional. In this case, dim(V)=dim(W).

Corollary 1. Let V be a finite-dimensional vector space with an ordered basis β, and let T:V→V be linear. Then T is invertible if and only if [T]β is invertible. Furthermore, [T−1]β=([T]β)−1.

Corollary 2. Let A be an n×n matrix. Then A is invertible if and only if LA is invertible. Furthermore, (LA)−1=LA−1.

V is isomorphic to W ⇔ dim(V)=dim(W)

Let W and W be finite-dimensional vector spaces (over the same field). Then V is isomorphic to W if and only if dim(V)=dim(W).

Corollary. Let V be a vector space over F. Then V is isomorphic to Fn if and only if dim(V)=n.

 ??

Let V and W be finite-dimensional vector spaces over F of dimensions n and m, respectively, and let β and γ be ordered bases for V and W, respectively. Then the function : (V,W)→Mm×n(F), defined by for T∈(V,W), is an isomorphism.

Corollary. Let V and W be finite-dimensional vector spaces of dimension n and m, respectively. Then (V,W) is finite-dimensional of dimension mn.

Φβ is an isomorphism

For any finite-dimensional vector space V with ordered basis β, Φβ is an isomorphism.

 ??

Let β and β' be two ordered bases for a finite-dimensional vector space V, and let . Then
(a) is invertible.
(b) For any V, .

[T]β'=Q-1[T]βQ

Let T be a linear operator on a finite-dimensional vector space V,and let β and β' be two ordered bases for V. Suppose that Q is the change of coordinate matrix that changes β'-coordinates into β-coordinates. Then

.

Corollary. Let A∈Mn×n(F), and le t γ be an ordered basis for Fn. Then [LA]γ=Q−1AQ, where Q is the n×n matrix whose jth column is the jth vector of γ.

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p(D)(x)=0 (p(D)∈C)⇒ x(k)exists (k∈N)

Any solution to a homogeneous linear differential equation with constant coefficients has derivatives of all orders; that is, if is a solution to such an equation, then exists for every positive integer k.

{solutions}= N(p(D))

The set of all solutions to a homogeneous linear differential equation with constant coefficients coincides with the null space of p(D), where p(t) is the auxiliary polynomial with the equation.

Corollary. The set of all solutions to s homogeneous linear differential equation with constant coefficients is a subspace of .

derivative of exponential function

For any exponential function .

{e-at} is a basis of N(p(D+aI))

The solution space for the differential equation,

is of dimension 1 and has as a basis.

Corollary. For any complex number c, the null space of the differential operator D-cI has {} as a basis.

is a solution

Let p(t) be the auxiliary polynomial for a homogeneous linear differential equation with constant coefficients. For any complex number c, if c is a zero of p(t), then to the differential equation.

dim(N(p(D)))=n

For any differential operator p(D) of order n, the null space of p(D) is an n_dimensional subspace of C.

Lemma 1. The differential operator D-cI: C to C is onto for any complex number c.

Lemma 2 Let V be a vector space, and suppose that T and U are linear operators on V such that U is onto and the null spaces of T and U are finite-dimensional, Then the null space of TU is finite-dimensional, and

dim(N(TU))=dim(N(U))+dim(N(U)).

Corollary. The solution space of any nth-order homogeneous linear differential equation with constant coefficients is an n-dimensional subspace of C.

ecit is linearly independent with each other (ci are distinct)

Given n distinct complex numbers , the set of exponential functions is linearly independent.

Corollary. For any nth-order homogeneous linear differential equation with constant coefficients, if the auxiliary polynomial has n distinct zeros , then is a basis for the solution space of the differential equation.

Lemma. For a given complex number c and positive integer n, suppose that (t-c)^n is athe auxiliary polynomial of a homogeneous linear differential equation with constant coefficients. Then the set

is a basis for the solution space of the equation.

general solution of homogeneous linear differential equation

Given a homogeneous linear differential equation with constant coefficients and auxiliary polynomial

where are positive integers and are distinct complex numbers, the following set is a basis for the solution space of the equation:

.

Elementary matrix operations and systems of linear equations

Elementary matrix operations

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Elementary matrix

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Rank of a matrix

The rank of a matrix A is the dimension of the column space of A.

Matrix inverses

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System of linear equations

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Determinants

If

is a 2×2 matrix with entries form a field F, then we define the determinant of A, denoted det(A) or |A|, to be the scalar .


*Theorem 1: linear function for a single row.
*Theorem 2: nonzero determinant ⇔ invertible matrix

Theorem 1: The function det: M2×2(F) → F is a linear function of each row of a 2×2 matrix when the other row is held fixed. That is, if and are in F² and is a scalar, then

and

Theorem 2: Let A M2×2(F). Then thee deter minant of A is nonzero if and only if A is invertible. Moreover, if A is invertible, then

Diagonalization

Characteristic polynomial of a linear operator/matrix

diagonalizable⇔basis of eigenvector

A linear operator T on a finite-dimensional vector space V is diagonalizable if and only if there exists an ordered basis β for V consisting of eigenvectors of T. Furthermore, if T is diagonalizable, is an ordered basis of eigenvectors of T, and D = [T]β then D is a diagonal matrix and is the eigenvalue corresponding to for .

eigenvalue⇔det(AIn)=0

Let A∈Mn×n(F). Then a scalar λ is an eigenvalue of A if and only if det(AIn)=0

characteristic polynomial

Let A∈Mn×n(F).
(a) The characteristic polynomial of A is a polynomial of degree n with leading coefficient(-1)n.
(b) A has at most n distinct eigenvalues.

υ to λ⇔υ∈N(T-λI)

Let T be a linear operator on a vector space V, and let λ be an eigenvalue of T.
A vector υ∈V is an eigenvector of T corresponding to λ if and only if υ≠0 and υ∈N(T-λI).

vi to λi⇔vi is linearly independent

Let T be a linear operator on a vector space V, and let be distinct eigenvalues of T. If are eigenvectors of t such that corresponds to (), then {} is linearly independent.

characteristic polynomial splits

The characteristic polynomial of any diagonalizable linear operator splits.

1 ≤ dim(Eλ) ≤ m

Let T be alinear operator on a finite-dimensional vectorspace V, and let λ be an eigenvalue of T having multiplicity . Then .

S = S1S2 ∪ ...∪ Sk is linearly independent

Let T be a linear operator on a vector space V, and let be distinct eigenvalues of T. For each let be a finite linearly independent subset of the eigenspace . Then is a linearly independent subset of V.

⇔T is diagonalizable

Let T be a linear operator on a finite-dimensional vector space V that the characteristic polynomial of T splits. Let be the distinct eigenvalues of T. Then
(a) T is diagonalizable if and only if the multiplicity of is equal to for all .
(b) If T is diagonalizable and is an ordered basis for for each , then is an ordered for V consisting of eigenvectors of T.

Test for diagonlization

Inner product spaces

Inner product, standard inner product on Fn, conjugate transpose, adjointTemplate:Dn, Frobenius inner product, complex/real inner product space, norm, length, conjugate linear, orthogonal, perpendicular, orthogonal, unit vector, orthonormal, normalization.

properties of linear product

Let V be an inner product space. Then for x,y,z\in V and c \in f, the following staements are true.
(a)
(b)
(c)
(d) if and only if
(e) If for all V, then .

law of norm

Let V be an inner product space over F. Then for all x,y\in V and c\in F, the following statements are true.
(a) .
(b) if and only if . In any case, .
(c)(Cauchy-Schwarz In equality).
(d)(Triangle Inequality).

orthonormal basis, Gram–Schmidt process, Fourier coefficients, orthogonal complement, orthogonal projection

span of orthogonal subset

Let V be an inner product space and be an orthogonal subset of V consisting of nonzero vectors. If ∈span(S), then

Gram-Schmidt process

Let V be an inner product space and S= be a linearly independent subset of V. DefineS'=, where and

Then S' is an orhtogonal set of nonzero vectors such that span(S')=span(S).

orthonormal basis

Let V be a nonzero finite-dimensional inner product space. Then V has an orthonormal basis β. Furthermore, if β = and x∈V, then

.

Corollary. Let V be a finite-dimensional inner product space with an orthonormal basis β =. Let T be a linear operator on V, and let A=[T]β. Then for any and , .

W by orthonormal basis

Let W be a finite-dimensional subspace of an inner product space V, and let ∈V. Then there exist unique vectors ∈W and ∈W such that . Furthermore, if is an orthornormal basis for W, then

.

S=\{v_1,v_2,\ldots,v_k\} Corollary. In the notation of Theorem 6.6, the vector is the unique vector in W that is "closest" to ; thet is, for any ∈W, , and this inequality is an equality if and onlly if .

properties of orthonormal set

Suppose that is an orthonormal set in an -dimensional inner product space V. Than
(a) S can be extended to an orthonormal basis for V.
(b) If W=span(S), then is an orhtonormal basis for W(using the preceding notation).
(c) If W is any subspace of V, then dim(V)=dim(W)+dim(W).

Least squares approximation, Minimal solutions to systems of linear equations

linear functional representation inner product

Let V be a finite-dimensional inner product space over F, and let :V→F be a linear transformation. Then there exists a unique vector ∈ V such that for all ∈ V.

definition of T*

Let V be a finite-dimensional inner product space, and let T be a linear operator on V. Then there exists a unique function T*:V→V such that for all ∈ V. Furthermore, T* is linear

[T*]β=[T]*β

Let V be a finite-dimensional inner product space, and let β be an orthonormal basis for V. If T is a linear operator on V, then

.

properties of T*

Let V be an inner product space, and let T and U be linear operators onV. Then
(a) (T+U)*=T*+U*;
(b) (T)*= T* for any c∈ F;
(c) (TU)*=U*T*;
(d) T**=T;
(e) I*=I.

Corollary. Let A and B be n×nmatrices. Then
(a) (A+B)*=A*+B*;
(b) (A)*= A* for any ∈ F;
(c) (AB)*=B*A*;
(d) A**=A;
(e) I*=I.

Least squares approximation

Let A ∈ Mm×n(F) and ∈Fm. Then there exists ∈ Fn such that and for all x∈ Fn

Lemma 1. let A ∈ Mm×n(F), ∈Fn, and ∈Fm. Then

Lemma 2. Let A ∈ Mm×n(F). Then rank(A*A)=rank(A).

Corollary.(of lemma 2) If A is an m×n matrix such that rank(A)=n, then A*A is invertible.

Minimal solutions to systems of linear equations

Let A ∈ Mm×n(F) and b∈ Fm. Suppose that is consistent. Then the following statements are true.
(a) There existes exactly one minimal solution of , and ∈R(LA*).
(b) The vector is the only solution to that lies in R(LA*); that is, if satisfies , then .

Canonical forms

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References

  • Linear Algebra 4th edition, by Stephen H. Friedberg Arnold J. Insel and Lawrence E. spence ISBN 7-04-016733-6
  • Linear Algebra 3rd edition, by Serge Lang (UTM) ISBN 0-387-96412-6
  • Linear Algebra and Its Applications 4th edition, by Gilbert Strang ISBN 0-03-010567-6