\(\newcommand{\footnotename}{footnote}\) \(\def \LWRfootnote {1}\) \(\newcommand {\footnote }[2][\LWRfootnote ]{{}^{\mathrm {#1}}}\) \(\newcommand {\footnotemark }[1][\LWRfootnote ]{{}^{\mathrm {#1}}}\) \(\let \LWRorighspace \hspace \) \(\renewcommand {\hspace }{\ifstar \LWRorighspace \LWRorighspace }\) \(\newcommand {\TextOrMath }[2]{#2}\) \(\newcommand {\mathnormal }[1]{{#1}}\) \(\newcommand \ensuremath [1]{#1}\) \(\newcommand {\LWRframebox }[2][]{\fbox {#2}} \newcommand {\framebox }[1][]{\LWRframebox } \) \(\newcommand {\setlength }[2]{}\) \(\newcommand {\addtolength }[2]{}\) \(\newcommand {\setcounter }[2]{}\) \(\newcommand {\addtocounter }[2]{}\) \(\newcommand {\arabic }[1]{}\) \(\newcommand {\number }[1]{}\) \(\newcommand {\noalign }[1]{\text {#1}\notag \\}\) \(\newcommand {\cline }[1]{}\) \(\newcommand {\directlua }[1]{\text {(directlua)}}\) \(\newcommand {\luatexdirectlua }[1]{\text {(directlua)}}\) \(\newcommand {\protect }{}\) \(\def \LWRabsorbnumber #1 {}\) \(\def \LWRabsorbquotenumber "#1 {}\) \(\newcommand {\LWRabsorboption }[1][]{}\) \(\newcommand {\LWRabsorbtwooptions }[1][]{\LWRabsorboption }\) \(\def \mathchar {\ifnextchar "\LWRabsorbquotenumber \LWRabsorbnumber }\) \(\def \mathcode #1={\mathchar }\) \(\let \delcode \mathcode \) \(\let \delimiter \mathchar \) \(\def \oe {\unicode {x0153}}\) \(\def \OE {\unicode {x0152}}\) \(\def \ae {\unicode {x00E6}}\) \(\def \AE {\unicode {x00C6}}\) \(\def \aa {\unicode {x00E5}}\) \(\def \AA {\unicode {x00C5}}\) \(\def \o {\unicode {x00F8}}\) \(\def \O {\unicode {x00D8}}\) \(\def \l {\unicode {x0142}}\) \(\def \L {\unicode {x0141}}\) \(\def \ss {\unicode {x00DF}}\) \(\def \SS {\unicode {x1E9E}}\) \(\def \dag {\unicode {x2020}}\) \(\def \ddag {\unicode {x2021}}\) \(\def \P {\unicode {x00B6}}\) \(\def \copyright {\unicode {x00A9}}\) \(\def \pounds {\unicode {x00A3}}\) \(\let \LWRref \ref \) \(\renewcommand {\ref }{\ifstar \LWRref \LWRref }\) \( \newcommand {\multicolumn }[3]{#3}\) \(\require {textcomp}\) \(\newcommand {\intertext }[1]{\text {#1}\notag \\}\) \(\let \Hat \hat \) \(\let \Check \check \) \(\let \Tilde \tilde \) \(\let \Acute \acute \) \(\let \Grave \grave \) \(\let \Dot \dot \) \(\let \Ddot \ddot \) \(\let \Breve \breve \) \(\let \Bar \bar \) \(\let \Vec \vec \) \(\newcommand {\bz }{\mathbf {0}}\) \(\newcommand {\bo }{\mathbf {1}}\) \(\newcommand {\id }{\operatorname {id}}\) \(\newcommand {\GL }{\operatorname {GL}}\) \(\newcommand {\im }{\operatorname {im}}\) \(\newcommand {\rank }{\operatorname {rank}}\) \(\newcommand {\sol }{\operatorname {sol}}\) \(\newcommand {\ann }{\operatorname {ann}}\) \(\newcommand {\rO }{\operatorname {O}}\) \(\newcommand {\rU }{\operatorname {U}}\) \(\newcommand {\rSU }{\operatorname {SU}}\) \(\newcommand {\ev }{\operatorname {ev}}\) \(\newcommand {\bil }{\operatorname {Bil}}\) \(\newcommand {\rad }{\operatorname {rad}}\) \(\newcommand {\Span }[1]{\operatorname {span}\{#1\}}\) \(\newcommand {\R }{\mathbb {R}}\) \(\newcommand {\C }{\mathbb {C}}\) \(\newcommand {\Z }{\mathbb {Z}}\) \(\newcommand {\F }{\mathbb {F}}\) \(\newcommand {\Q }{\mathbb {Q}}\) \(\newcommand {\N }{\mathbb {N}}\) \(\renewcommand {\P }{\mathbb {P}}\) \(\newcommand {\I }{\mathrm {I}}\) \(\newcommand {\half }{\tfrac 12}\) \(\newcommand {\rel }{\mathrel {\mathrm {rel}}}\) \(\renewcommand {\vec }[3]{(#1_{#2},\dots ,#1_{#3})}\) \(\newcommand {\lst }[3]{#1_{#2},\dots ,#1_{#3}}\) \(\newcommand {\plst }[3]{#1_{#2}+\dots +#1_{#3}}\) \(\newcommand {\oplst }[3]{#1_{#2}\oplus \dots \oplus #1_{#3}}\) \(\newcommand {\pplst }[3]{#1_{#2}\times \dots \times #1_{#3}}\) \(\newcommand {\dlst }[3]{\lst {#1^{*}}{#2}{#3}}\) \(\newcommand {\dlc }[4]{\lc #1{#2^{*}}#3#4}\) \(\newcommand {\hmg }[3]{[#1_{#2},\dots ,#1_{#3}]}\) \(\newcommand {\rng }[2]{#1,\dots ,#2}\) \(\newcommand {\lc }[4]{#1_{#3}#2_{#3}+\dots +#1_{#4}#2_{#4}}\) \(\newcommand {\plus }[2]{#1+\dots +#2}\) \(\newcommand {\set }[1]{\{#1\}}\) \(\newcommand {\abs }[1]{\lvert #1\rvert }\) \(\newcommand {\ip }[1]{\langle #1\rangle }\) \(\newcommand {\norm }[1]{\|#1\|}\) \(\newcommand {\bx }{\mathbf {x}}\) \(\newcommand {\be }{\mathbf {e}}\) \(\newcommand {\bq }{\mathbf {q}}\) \(\newcommand {\bu }{\mathbf {u}}\) \(\newcommand {\by }{\mathbf {y}}\) \(\newcommand {\bv }{\mathbf {v}}\) \(\newcommand {\E }{\mathbb {E}}\) \(\newcommand {\cI }{\mathcal {I}}\) \(\newcommand {\cB }{\mathcal {B}}\) \(\newcommand {\sub }{\subseteq }\) \(\newcommand {\st }{\mathrel {|}}\) \(\newcommand {\bw }[3]{#1\leq #2\leq #3}\) \(\newcommand {\col }[3]{(#1_{#2})_{#2\in #3}}\) \(\newcommand {\supp }{\mathrm {supp}}\) \(\newcommand {\restr }[1]{_{|#1}}\) \(\newcommand {\re }{\operatorname {Re}}\)

MA22020: Exercise sheet 2

    Warmup questions

  • 1. Let \(U_1,U_2,U_3\leq \R ^3\) be the \(1\)-dimensional subspaces spanned by \((1,2,0)\), \((1,1,1)\) and \((2,3,1)\) respectively.

    Which of the following sums are direct?

    • (a) \(U_i+U_j\), for \(1\leq i<j\leq 3\).

    • (b) \(U_1+U_2+U_3\).

  • 2. Let \(V_i\leq V\), for \(\bw 1ik\). Prove the converse of Corollary 2.8: if

    \begin{equation*} \dim \plst {V}1k=\plst {\dim V}1k \end{equation*}

    then the sum \(\plst {V}1k\) is direct.

  • 3. Let \(U\leq V\). Show that congruence modulo \(U\) is an equivalence relation.

  • 4. Let \(U=\Span {(1,-1,0),(0,1,-1)}\leq \R ^{3}\). Determine which, if any, of the following cosets are equal:

    \begin{equation*} (1,2,3)+U,\qquad (3,3,0)+U,\qquad (1,1,1)+U. \end{equation*}

  • 5. Let \(U\leq V\) and \(q:V\to V/U\) the quotient map. Let \(W\) be a complement to \(U\).

    Show that \(q_{|W}:W\to V/U\) is an isomorphism.

    Homework

  • 6. Let \(V\) be a vector space. A linear map \(\pi :V\to V\) is called a projection if \(\pi \circ \pi =\pi \).

    In this case, prove that \(\ker \pi \cap \im \pi =\set {0}\) and deduce that \(V=\ker \pi \oplus \im \pi \).

  • 7. Let \(U,W\leq V\). Define a linear map \(\phi :U\to (U+W)/W\) by \(\phi (u)=u+W\).

    • (a) Use the first isomorphism theorem, applied to \(\phi \), to prove the second isomorphism theorem:

      \begin{equation*} U/(U\cap W)\cong (U+W)/W. \end{equation*}

    • (b) Deduce that, when \(V\) is finite-dimensional,

      \begin{equation*} \dim (U+W)=\dim U+\dim W-\dim (U\cap W). \end{equation*}

Please hand in at 4W level 1 by NOON on Thursday 30th October 2025