\(\newcommand{\footnotename}{footnote}\)
\(\def \LWRfootnote {1}\)
\(\newcommand {\footnote }[2][\LWRfootnote ]{{}^{\mathrm {#1}}}\)
\(\newcommand {\footnotemark }[1][\LWRfootnote ]{{}^{\mathrm {#1}}}\)
\(\let \LWRorighspace \hspace \)
\(\renewcommand {\hspace }{\ifstar \LWRorighspace \LWRorighspace }\)
\(\newcommand {\mathnormal }[1]{{#1}}\)
\(\newcommand \ensuremath [1]{#1}\)
\(\newcommand {\LWRframebox }[2][]{\fbox {#2}} \newcommand {\framebox }[1][]{\LWRframebox } \)
\(\newcommand {\setlength }[2]{}\)
\(\newcommand {\addtolength }[2]{}\)
\(\newcommand {\setcounter }[2]{}\)
\(\newcommand {\addtocounter }[2]{}\)
\(\newcommand {\arabic }[1]{}\)
\(\newcommand {\number }[1]{}\)
\(\newcommand {\noalign }[1]{\text {#1}\notag \\}\)
\(\newcommand {\cline }[1]{}\)
\(\newcommand {\directlua }[1]{\text {(directlua)}}\)
\(\newcommand {\luatexdirectlua }[1]{\text {(directlua)}}\)
\(\newcommand {\protect }{}\)
\(\def \LWRabsorbnumber #1 {}\)
\(\def \LWRabsorbquotenumber "#1 {}\)
\(\newcommand {\LWRabsorboption }[1][]{}\)
\(\newcommand {\LWRabsorbtwooptions }[1][]{\LWRabsorboption }\)
\(\def \mathchar {\ifnextchar "\LWRabsorbquotenumber \LWRabsorbnumber }\)
\(\def \mathcode #1={\mathchar }\)
\(\let \delcode \mathcode \)
\(\let \delimiter \mathchar \)
\(\def \oe {\unicode {x0153}}\)
\(\def \OE {\unicode {x0152}}\)
\(\def \ae {\unicode {x00E6}}\)
\(\def \AE {\unicode {x00C6}}\)
\(\def \aa {\unicode {x00E5}}\)
\(\def \AA {\unicode {x00C5}}\)
\(\def \o {\unicode {x00F8}}\)
\(\def \O {\unicode {x00D8}}\)
\(\def \l {\unicode {x0142}}\)
\(\def \L {\unicode {x0141}}\)
\(\def \ss {\unicode {x00DF}}\)
\(\def \SS {\unicode {x1E9E}}\)
\(\def \dag {\unicode {x2020}}\)
\(\def \ddag {\unicode {x2021}}\)
\(\def \P {\unicode {x00B6}}\)
\(\def \copyright {\unicode {x00A9}}\)
\(\def \pounds {\unicode {x00A3}}\)
\(\let \LWRref \ref \)
\(\renewcommand {\ref }{\ifstar \LWRref \LWRref }\)
\( \newcommand {\multicolumn }[3]{#3}\)
\(\require {textcomp}\)
\(\newcommand {\intertext }[1]{\text {#1}\notag \\}\)
\(\let \Hat \hat \)
\(\let \Check \check \)
\(\let \Tilde \tilde \)
\(\let \Acute \acute \)
\(\let \Grave \grave \)
\(\let \Dot \dot \)
\(\let \Ddot \ddot \)
\(\let \Breve \breve \)
\(\let \Bar \bar \)
\(\let \Vec \vec \)
\(\require {mathtools}\)
\(\newenvironment {crampedsubarray}[1]{}{}\)
\(\newcommand {\smashoperator }[2][]{#2\limits }\)
\(\newcommand {\SwapAboveDisplaySkip }{}\)
\(\newcommand {\LaTeXunderbrace }[1]{\underbrace {#1}}\)
\(\newcommand {\LaTeXoverbrace }[1]{\overbrace {#1}}\)
\(\newcommand {\LWRmultlined }[1][]{\begin {multline*}}\)
\(\newenvironment {multlined}[1][]{\LWRmultlined }{\end {multline*}}\)
\(\let \LWRorigshoveleft \shoveleft \)
\(\renewcommand {\shoveleft }[1][]{\LWRorigshoveleft }\)
\(\let \LWRorigshoveright \shoveright \)
\(\renewcommand {\shoveright }[1][]{\LWRorigshoveright }\)
\(\newcommand {\shortintertext }[1]{\text {#1}\notag \\}\)
\(\newcommand {\vcentcolon }{\mathrel {\unicode {x2236}}}\)
\(\newcommand {\bigzero }{\smash {\text {\huge 0}}}\)
\(\newcommand {\am }{\mathrm {am}}\)
\(\newcommand {\gm }{\mathrm {gm}}\)
\(\newcommand {\id }{\operatorname {id}}\)
\(\newcommand {\GL }{\operatorname {GL}}\)
\(\newcommand {\im }{\operatorname {im}}\)
\(\newcommand {\rank }{\operatorname {rank}}\)
\(\newcommand {\sol }{\operatorname {sol}}\)
\(\newcommand {\ann }{\operatorname {ann}}\)
\(\newcommand {\rO }{\operatorname {O}}\)
\(\newcommand {\rU }{\operatorname {U}}\)
\(\newcommand {\rSU }{\operatorname {SU}}\)
\(\newcommand {\ev }{\operatorname {ev}}\)
\(\newcommand {\bil }{\operatorname {Bil}}\)
\(\newcommand {\rad }{\operatorname {rad}}\)
\(\newcommand {\Span }[1]{\operatorname {span}\{#1\}}\)
\(\newcommand {\R }{\mathbb {R}}\)
\(\newcommand {\C }{\mathbb {C}}\)
\(\newcommand {\Z }{\mathbb {Z}}\)
\(\newcommand {\F }{\mathbb {F}}\)
\(\newcommand {\Q }{\mathbb {Q}}\)
\(\newcommand {\N }{\mathbb {N}}\)
\(\renewcommand {\P }{\mathbb {P}}\)
\(\newcommand {\I }{\mathrm {I}}\)
\(\newcommand {\half }{\tfrac 12}\)
\(\newcommand {\rel }{\mathrel {\mathrm {rel}}}\)
\(\renewcommand {\vec }[3]{(#1_{#2},\dots ,#1_{#3})}\)
\(\newcommand {\lst }[3]{#1_{#2},\dots ,#1_{#3}}\)
\(\newcommand {\plst }[3]{#1_{#2}+\dots +#1_{#3}}\)
\(\newcommand {\oplst }[3]{#1_{#2}\oplus \dots \oplus #1_{#3}}\)
\(\newcommand {\pplst }[3]{#1_{#2}\times \dots \times #1_{#3}}\)
\(\newcommand {\dlst }[3]{\lst {#1^{*}}{#2}{#3}}\)
\(\newcommand {\dlc }[4]{\lc #1{#2^{*}}#3#4}\)
\(\newcommand {\hmg }[3]{[#1_{#2},\dots ,#1_{#3}]}\)
\(\newcommand {\rng }[2]{#1,\dots ,#2}\)
\(\newcommand {\lc }[4]{#1_{#3}#2_{#3}+\dots +#1_{#4}#2_{#4}}\)
\(\newcommand {\plus }[2]{#1+\dots +#2}\)
\(\newcommand {\set }[1]{\{#1\}}\)
\(\newcommand {\abs }[1]{\lvert #1\rvert }\)
\(\newcommand {\ip }[1]{\langle #1\rangle }\)
\(\newcommand {\norm }[1]{\|#1\|}\)
\(\newcommand {\bx }{\mathbf {x}}\)
\(\newcommand {\be }{\mathbf {e}}\)
\(\newcommand {\bq }{\mathbf {q}}\)
\(\newcommand {\bu }{\mathbf {u}}\)
\(\newcommand {\by }{\mathbf {y}}\)
\(\newcommand {\bv }{\mathbf {v}}\)
\(\newcommand {\E }{\mathbb {E}}\)
\(\newcommand {\cI }{\mathcal {I}}\)
\(\newcommand {\cB }{\mathcal {B}}\)
\(\newcommand {\sub }{\subseteq }\)
\(\newcommand {\st }{\mathrel {|}}\)
\(\newcommand {\bw }[3]{#1\leq #2\leq #3}\)
\(\newcommand {\col }[3]{(#1_{#2})_{#2\in #3}}\)
\(\newcommand {\supp }{\mathrm {supp}}\)
\(\newcommand {\restr }[1]{_{|#1}}\)
Chapter 2 Sums and quotients
We will discuss various ways of building new vector spaces out of old ones.
2.1 Sums of subspaces
\(\plst {V}1k\) is the smallest subspace of \(V\) that contains each \(V_i\). More precisely:
-
Proof. It suffices to prove (2) since (1) then follows by taking \(W=V\).
For (2), first note that \(\plst {V}1k\) is a subset of \(W\): if \(v_i\in V_i\) then \(v_i\in W\) so that \(\plst {v}1k\in W\) since \(W\) is closed under addition.
Now observe that each \(V_i\leq \plst {V}1k\) since we can write any \(v_i\in V_i\) as \(0+\dots +v_i+\dots +0\in \plst {V}1k\). In particular, \(0\in \plst {V}1k\).
Finally, we show that \(\plst {V}1k\) is a subspace. If \(\plst {v}1k,\plst {w}1k\in \plst {V}1k\), with \(v_i,w_i\in V_i\), for all \(i\), and \(\lambda \in \F \) then
\(\seteqnumber{0}{2.}{0}\)\begin{equation*} (\plst {v}1k)+\lambda (\plst {w}1k)=\plus {(v_1+\lambda w_1)}{(v_k+\lambda w_k)}\in \plst {V}1k \end{equation*}
since each \(v_i+\lambda w_i\in V_i\). □