\(\newcommand{\footnotename}{footnote}\)
\(\def \LWRfootnote {1}\)
\(\newcommand {\footnote }[2][\LWRfootnote ]{{}^{\mathrm {#1}}}\)
\(\newcommand {\footnotemark }[1][\LWRfootnote ]{{}^{\mathrm {#1}}}\)
\(\let \LWRorighspace \hspace \)
\(\renewcommand {\hspace }{\ifstar \LWRorighspace \LWRorighspace }\)
\(\newcommand {\mathnormal }[1]{{#1}}\)
\(\newcommand \ensuremath [1]{#1}\)
\(\newcommand {\LWRframebox }[2][]{\fbox {#2}} \newcommand {\framebox }[1][]{\LWRframebox } \)
\(\newcommand {\setlength }[2]{}\)
\(\newcommand {\addtolength }[2]{}\)
\(\newcommand {\setcounter }[2]{}\)
\(\newcommand {\addtocounter }[2]{}\)
\(\newcommand {\arabic }[1]{}\)
\(\newcommand {\number }[1]{}\)
\(\newcommand {\noalign }[1]{\text {#1}\notag \\}\)
\(\newcommand {\cline }[1]{}\)
\(\newcommand {\directlua }[1]{\text {(directlua)}}\)
\(\newcommand {\luatexdirectlua }[1]{\text {(directlua)}}\)
\(\newcommand {\protect }{}\)
\(\def \LWRabsorbnumber #1 {}\)
\(\def \LWRabsorbquotenumber "#1 {}\)
\(\newcommand {\LWRabsorboption }[1][]{}\)
\(\newcommand {\LWRabsorbtwooptions }[1][]{\LWRabsorboption }\)
\(\def \mathchar {\ifnextchar "\LWRabsorbquotenumber \LWRabsorbnumber }\)
\(\def \mathcode #1={\mathchar }\)
\(\let \delcode \mathcode \)
\(\let \delimiter \mathchar \)
\(\let \LWRref \ref \)
\(\renewcommand {\ref }{\ifstar \LWRref \LWRref }\)
\(\newcommand {\intertext }[1]{\text {#1}\notag \\}\)
\(\let \Hat \hat \)
\(\let \Check \check \)
\(\let \Tilde \tilde \)
\(\let \Acute \acute \)
\(\let \Grave \grave \)
\(\let \Dot \dot \)
\(\let \Ddot \ddot \)
\(\let \Breve \breve \)
\(\let \Bar \bar \)
\(\let \Vec \vec \)
\( \newcommand {\multicolumn }[3]{#3}\)
\(\newcommand {\mathllap }[2][]{{#1#2}}\)
\(\newcommand {\mathrlap }[2][]{{#1#2}}\)
\(\newcommand {\mathclap }[2][]{{#1#2}}\)
\(\newcommand {\mathmbox }[1]{#1}\)
\(\newcommand {\clap }[1]{#1}\)
\(\newcommand {\LWRmathmakebox }[2][]{#2}\)
\(\newcommand {\mathmakebox }[1][]{\LWRmathmakebox }\)
\(\newcommand {\cramped }[2][]{{#1#2}}\)
\(\newcommand {\crampedllap }[2][]{{#1#2}}\)
\(\newcommand {\crampedrlap }[2][]{{#1#2}}\)
\(\newcommand {\crampedclap }[2][]{{#1#2}}\)
\(\newenvironment {crampedsubarray}[1]{}{}\)
\(\newcommand {\crampedsubstack }{}\)
\(\newcommand {\smashoperator }[2][]{#2\limits }\)
\(\newcommand {\adjustlimits }{}\)
\(\newcommand {\SwapAboveDisplaySkip }{}\)
\(\require {extpfeil}\)
\(\Newextarrow \xleftrightarrow {10,10}{0x2194}\)
\(\Newextarrow \xLeftarrow {10,10}{0x21d0}\)
\(\Newextarrow \xhookleftarrow {10,10}{0x21a9}\)
\(\Newextarrow \xmapsto {10,10}{0x21a6}\)
\(\Newextarrow \xRightarrow {10,10}{0x21d2}\)
\(\Newextarrow \xLeftrightarrow {10,10}{0x21d4}\)
\(\Newextarrow \xhookrightarrow {10,10}{0x21aa}\)
\(\Newextarrow \xrightharpoondown {10,10}{0x21c1}\)
\(\Newextarrow \xleftharpoondown {10,10}{0x21bd}\)
\(\Newextarrow \xrightleftharpoons {10,10}{0x21cc}\)
\(\Newextarrow \xrightharpoonup {10,10}{0x21c0}\)
\(\Newextarrow \xleftharpoonup {10,10}{0x21bc}\)
\(\Newextarrow \xleftrightharpoons {10,10}{0x21cb}\)
\(\newcommand {\LWRdounderbracket }[3]{\mathinner {\underset {#3}{\underline {\llcorner {#1}\lrcorner }}}}\)
\(\newcommand {\LWRunderbracket }[2][]{\LWRdounderbracket {#2}}\)
\(\newcommand {\underbracket }[1][]{\LWRunderbracket }\)
\(\newcommand {\LWRdooverbracket }[3]{\mathinner {\overset {#3}{\overline {\ulcorner {#1}\urcorner }}}}\)
\(\newcommand {\LWRoverbracket }[2][]{\LWRdooverbracket {#2}}\)
\(\newcommand {\overbracket }[1][]{\LWRoverbracket }\)
\(\newcommand {\LaTeXunderbrace }[1]{\underbrace {#1}}\)
\(\newcommand {\LaTeXoverbrace }[1]{\overbrace {#1}}\)
\(\newenvironment {matrix*}[1][]{\begin {matrix}}{\end {matrix}}\)
\(\newenvironment {pmatrix*}[1][]{\begin {pmatrix}}{\end {pmatrix}}\)
\(\newenvironment {bmatrix*}[1][]{\begin {bmatrix}}{\end {bmatrix}}\)
\(\newenvironment {Bmatrix*}[1][]{\begin {Bmatrix}}{\end {Bmatrix}}\)
\(\newenvironment {vmatrix*}[1][]{\begin {vmatrix}}{\end {vmatrix}}\)
\(\newenvironment {Vmatrix*}[1][]{\begin {Vmatrix}}{\end {Vmatrix}}\)
\(\newenvironment {smallmatrix*}[1][]{\begin {matrix}}{\end {matrix}}\)
\(\newenvironment {psmallmatrix*}[1][]{\begin {pmatrix}}{\end {pmatrix}}\)
\(\newenvironment {bsmallmatrix*}[1][]{\begin {bmatrix}}{\end {bmatrix}}\)
\(\newenvironment {Bsmallmatrix*}[1][]{\begin {Bmatrix}}{\end {Bmatrix}}\)
\(\newenvironment {vsmallmatrix*}[1][]{\begin {vmatrix}}{\end {vmatrix}}\)
\(\newenvironment {Vsmallmatrix*}[1][]{\begin {Vmatrix}}{\end {Vmatrix}}\)
\(\newenvironment {psmallmatrix}[1][]{\begin {pmatrix}}{\end {pmatrix}}\)
\(\newenvironment {bsmallmatrix}[1][]{\begin {bmatrix}}{\end {bmatrix}}\)
\(\newenvironment {Bsmallmatrix}[1][]{\begin {Bmatrix}}{\end {Bmatrix}}\)
\(\newenvironment {vsmallmatrix}[1][]{\begin {vmatrix}}{\end {vmatrix}}\)
\(\newenvironment {Vsmallmatrix}[1][]{\begin {Vmatrix}}{\end {Vmatrix}}\)
\(\newcommand {\LWRmultlined }[1][]{\begin {multline*}}\)
\(\newenvironment {multlined}[1][]{\LWRmultlined }{\end {multline*}}\)
\(\let \LWRorigshoveleft \shoveleft \)
\(\renewcommand {\shoveleft }[1][]{\LWRorigshoveleft }\)
\(\let \LWRorigshoveright \shoveright \)
\(\renewcommand {\shoveright }[1][]{\LWRorigshoveright }\)
\(\newenvironment {dcases}{\begin {cases}}{\end {cases}}\)
\(\newenvironment {dcases*}{\begin {cases}}{\end {cases}}\)
\(\newenvironment {rcases}{\begin {cases}}{\end {cases}}\)
\(\newenvironment {rcases*}{\begin {cases}}{\end {cases}}\)
\(\newenvironment {drcases}{\begin {cases}}{\end {cases}}\)
\(\newenvironment {drcases*}{\begin {cases}}{\end {cases}}\)
\(\newenvironment {cases*}{\begin {cases}}{\end {cases}}\)
\(\newcommand {\MoveEqLeft }[1][]{}\)
\(\def \LWRAboxed #1!|!{\fbox {\(#1\)}&\fbox {\(#2\)}} \newcommand {\Aboxed }[1]{\LWRAboxed #1&&!|!} \)
\( \newcommand {\LWRABLines }[1][\Updownarrow ]{#1 \notag \\}\newcommand {\ArrowBetweenLines }{\ifstar \LWRABLines \LWRABLines } \)
\(\newcommand {\shortintertext }[1]{\text {#1}\notag \\}\)
\(\newcommand {\vdotswithin }[1]{\hspace {.5em}\vdots }\)
\(\newcommand {\LWRshortvdotswithinstar }[1]{\vdots \hspace {.5em} & \\}\)
\(\newcommand {\LWRshortvdotswithinnostar }[1]{& \hspace {.5em}\vdots \\}\)
\(\newcommand {\shortvdotswithin }{\ifstar \LWRshortvdotswithinstar \LWRshortvdotswithinnostar }\)
\(\newcommand {\MTFlushSpaceAbove }{}\)
\(\newcommand {\MTFlushSpaceBelow }{\\}\)
\(\newcommand \lparen {(}\)
\(\newcommand \rparen {)}\)
\(\newcommand {\ordinarycolon }{:}\)
\(\newcommand {\vcentcolon }{\mathrel {\unicode {x2236}}}\)
\(\newcommand \dblcolon {\mathrel {\unicode {x2237}}}\)
\(\newcommand \coloneqq {\mathrel {\unicode {x2236}\!=}}\)
\(\newcommand \Coloneqq {\mathrel {\unicode {x2237}\!=}}\)
\(\newcommand \coloneq {\mathrel {\unicode {x2236}-}}\)
\(\newcommand \Coloneq {\mathrel {\unicode {x2237}-}}\)
\(\newcommand \eqqcolon {\mathrel {=\!\unicode {x2236}}}\)
\(\newcommand \Eqqcolon {\mathrel {=\!\unicode {x2237}}}\)
\(\newcommand \eqcolon {\mathrel {-\unicode {x2236}}}\)
\(\newcommand \Eqcolon {\mathrel {-\unicode {x2237}}}\)
\(\newcommand \colonapprox {\mathrel {\unicode {x2236}\!\approx }}\)
\(\newcommand \Colonapprox {\mathrel {\unicode {x2237}\!\approx }}\)
\(\newcommand \colonsim {\mathrel {\unicode {x2236}\!\sim }}\)
\(\newcommand \Colonsim {\mathrel {\unicode {x2237}\!\sim }}\)
\(\newcommand {\nuparrow }{\mathrel {\cancel {\uparrow }}}\)
\(\newcommand {\ndownarrow }{\mathrel {\cancel {\downarrow }}}\)
\(\newcommand {\bigtimes }{\mathop {\Large \times }\limits }\)
\(\newcommand {\prescript }[3]{{}^{#1}_{#2}#3}\)
\(\newenvironment {lgathered}{\begin {gathered}}{\end {gathered}}\)
\(\newenvironment {rgathered}{\begin {gathered}}{\end {gathered}}\)
\(\newcommand {\splitfrac }[2]{{}^{#1}_{#2}}\)
\(\let \splitdfrac \splitfrac \)
\(\newcommand {\bigzero }{\smash {\text {\huge 0}}}\)
\(\newcommand {\am }{\mathrm {am}}\)
\(\newcommand {\gm }{\mathrm {gm}}\)
\(\newcommand {\id }{\operatorname {id}}\)
\(\newcommand {\GL }{\operatorname {GL}}\)
\(\newcommand {\im }{\operatorname {im}}\)
\(\newcommand {\rank }{\operatorname {rank}}\)
\(\newcommand {\sol }{\operatorname {sol}}\)
\(\newcommand {\ann }{\operatorname {ann}}\)
\(\newcommand {\rO }{\operatorname {O}}\)
\(\newcommand {\rU }{\operatorname {U}}\)
\(\newcommand {\rSU }{\operatorname {SU}}\)
\(\newcommand {\ev }{\operatorname {ev}}\)
\(\newcommand {\bil }{\operatorname {Bil}}\)
\(\newcommand {\rad }{\operatorname {rad}}\)
\(\newcommand {\Span }[1]{\operatorname {span}\{#1\}}\)
\(\newcommand {\R }{\mathbb {R}}\)
\(\newcommand {\C }{\mathbb {C}}\)
\(\newcommand {\Z }{\mathbb {Z}}\)
\(\newcommand {\F }{\mathbb {F}}\)
\(\newcommand {\Q }{\mathbb {Q}}\)
\(\newcommand {\N }{\mathbb {N}}\)
\(\renewcommand {\P }{\mathbb {P}}\)
\(\newcommand {\I }{\mathrm {I}}\)
\(\newcommand {\half }{\tfrac 12}\)
\(\newcommand {\rel }{\mathrel {\mathrm {rel}}}\)
\(\renewcommand {\vec }[3]{(#1_{#2},\dots ,#1_{#3})}\)
\(\newcommand {\lst }[3]{#1_{#2},\dots ,#1_{#3}}\)
\(\newcommand {\plst }[3]{#1_{#2}+\dots +#1_{#3}}\)
\(\newcommand {\oplst }[3]{#1_{#2}\oplus \dots \oplus #1_{#3}}\)
\(\newcommand {\pplst }[3]{#1_{#2}\times \dots \times #1_{#3}}\)
\(\newcommand {\dlst }[3]{\lst {#1^{*}}{#2}{#3}}\)
\(\newcommand {\dlc }[4]{\lc #1{#2^{*}}#3#4}\)
\(\newcommand {\hmg }[3]{[#1_{#2},\dots ,#1_{#3}]}\)
\(\newcommand {\rng }[2]{#1,\dots ,#2}\)
\(\newcommand {\lc }[4]{#1_{#3}#2_{#3}+\dots +#1_{#4}#2_{#4}}\)
\(\newcommand {\plus }[2]{#1+\dots +#2}\)
\(\newcommand {\set }[1]{\{#1\}}\)
\(\newcommand {\abs }[1]{\lvert #1\rvert }\)
\(\newcommand {\ip }[1]{\langle #1\rangle }\)
\(\newcommand {\norm }[1]{\|#1\|}\)
\(\newcommand {\bx }{\mathbf {x}}\)
\(\newcommand {\be }{\mathbf {e}}\)
\(\newcommand {\bq }{\mathbf {q}}\)
\(\newcommand {\bu }{\mathbf {u}}\)
\(\newcommand {\by }{\mathbf {y}}\)
\(\newcommand {\bv }{\mathbf {v}}\)
\(\newcommand {\E }{\mathbb {E}}\)
\(\newcommand {\cI }{\mathcal {I}}\)
\(\newcommand {\cB }{\mathcal {B}}\)
\(\newcommand {\sub }{\subseteq }\)
\(\newcommand {\st }{\mathrel {|}}\)
\(\newcommand {\bw }[3]{#1\leq #2\leq #3}\)
\(\newcommand {\col }[3]{(#1_{#2})_{#2\in #3}}\)
\(\newcommand {\supp }{\mathrm {supp}}\)
\(\newcommand {\restr }[1]{_{|#1}}\)
1.3 Bases
-
Definitions. Let \(\lst {v}1n\) be a list of vectors in a vector space \(V\).
-
(1) The span of \(\lst {v}1n\) is
\(\seteqnumber{0}{1.}{0}\)\begin{equation*} \Span {\lst {v}1n}:=\set {\lc \lambda {v}1n\st \lambda _i\in \F , 1\leq i\leq n}\leq V. \end{equation*}
-
(2) \(\lst {v}1n\) span \(V\) (or are a spanning list for \(V\)) if \(\Span {\lst {v}1n}=V\).
-
(3) \(\lst {v}1n\) are linearly independent if, whenever \(\lc \lambda {v}1n=0\), then each \(\lambda _i=0\), \(1\leq i\leq n\), and linearly dependent otherwise.
-
(4) \(\lst {v}1n\) is a basis for \(V\) if they are linearly independent and span \(V\).
-
-
Proposition 1.1 (Algebra 1B, Proposition 1.3.4). \(\lst {v}1n\) is a basis for \(V\) if and only if any \(v\in V\) can be written in the form
\(\seteqnumber{0}{1.}{0}\)\begin{equation} \label {eq:1} v=\lc \lambda {v}1n \end{equation}
for unique \(\lst \lambda 1n\in \F \). In this case, \(\vec \lambda 1n\) is called the coordinate vector of \(v\) with respect to \(\lst {v}1n\).
1.3.1 Standard bases
-
Proposition 1.2. For \(\cI \) a set and \(i\in \cI \), define \(e_i\in \F ^{\cI }\) by
\(\seteqnumber{0}{1.}{1}\)\begin{equation*} e_i(j)= \begin{cases} 1&\text {if $i=j$}\\0&\text {if $i\neq j$}, \end {cases} \end{equation*}
for all \(j\in \cI \).
If \(\cI \) is finite then \((e_i)_{i\in \cI }\) is a basis, called the standard basis, of \(\F ^{\cI }\).
In particular, \(\dim \F ^{\cI }=\abs {\cI }\).