\(\newcommand{\footnotename}{footnote}\)
\(\def \LWRfootnote {1}\)
\(\newcommand {\footnote }[2][\LWRfootnote ]{{}^{\mathrm {#1}}}\)
\(\newcommand {\footnotemark }[1][\LWRfootnote ]{{}^{\mathrm {#1}}}\)
\(\let \LWRorighspace \hspace \)
\(\renewcommand {\hspace }{\ifstar \LWRorighspace \LWRorighspace }\)
\(\newcommand {\TextOrMath }[2]{#2}\)
\(\newcommand {\mathnormal }[1]{{#1}}\)
\(\newcommand \ensuremath [1]{#1}\)
\(\newcommand {\LWRframebox }[2][]{\fbox {#2}} \newcommand {\framebox }[1][]{\LWRframebox } \)
\(\newcommand {\setlength }[2]{}\)
\(\newcommand {\addtolength }[2]{}\)
\(\newcommand {\setcounter }[2]{}\)
\(\newcommand {\addtocounter }[2]{}\)
\(\newcommand {\arabic }[1]{}\)
\(\newcommand {\number }[1]{}\)
\(\newcommand {\noalign }[1]{\text {#1}\notag \\}\)
\(\newcommand {\cline }[1]{}\)
\(\newcommand {\directlua }[1]{\text {(directlua)}}\)
\(\newcommand {\luatexdirectlua }[1]{\text {(directlua)}}\)
\(\newcommand {\protect }{}\)
\(\def \LWRabsorbnumber #1 {}\)
\(\def \LWRabsorbquotenumber "#1 {}\)
\(\newcommand {\LWRabsorboption }[1][]{}\)
\(\newcommand {\LWRabsorbtwooptions }[1][]{\LWRabsorboption }\)
\(\def \mathchar {\ifnextchar "\LWRabsorbquotenumber \LWRabsorbnumber }\)
\(\def \mathcode #1={\mathchar }\)
\(\let \delcode \mathcode \)
\(\let \delimiter \mathchar \)
\(\def \oe {\unicode {x0153}}\)
\(\def \OE {\unicode {x0152}}\)
\(\def \ae {\unicode {x00E6}}\)
\(\def \AE {\unicode {x00C6}}\)
\(\def \aa {\unicode {x00E5}}\)
\(\def \AA {\unicode {x00C5}}\)
\(\def \o {\unicode {x00F8}}\)
\(\def \O {\unicode {x00D8}}\)
\(\def \l {\unicode {x0142}}\)
\(\def \L {\unicode {x0141}}\)
\(\def \ss {\unicode {x00DF}}\)
\(\def \SS {\unicode {x1E9E}}\)
\(\def \dag {\unicode {x2020}}\)
\(\def \ddag {\unicode {x2021}}\)
\(\def \P {\unicode {x00B6}}\)
\(\def \copyright {\unicode {x00A9}}\)
\(\def \pounds {\unicode {x00A3}}\)
\(\let \LWRref \ref \)
\(\renewcommand {\ref }{\ifstar \LWRref \LWRref }\)
\( \newcommand {\multicolumn }[3]{#3}\)
\(\require {textcomp}\)
\(\newcommand {\intertext }[1]{\text {#1}\notag \\}\)
\(\let \Hat \hat \)
\(\let \Check \check \)
\(\let \Tilde \tilde \)
\(\let \Acute \acute \)
\(\let \Grave \grave \)
\(\let \Dot \dot \)
\(\let \Ddot \ddot \)
\(\let \Breve \breve \)
\(\let \Bar \bar \)
\(\let \Vec \vec \)
\(\newcommand {\bz }{\mathbf {0}}\)
\(\newcommand {\bo }{\mathbf {1}}\)
\(\newcommand {\id }{\operatorname {id}}\)
\(\newcommand {\GL }{\operatorname {GL}}\)
\(\newcommand {\im }{\operatorname {im}}\)
\(\newcommand {\rank }{\operatorname {rank}}\)
\(\newcommand {\sol }{\operatorname {sol}}\)
\(\newcommand {\ann }{\operatorname {ann}}\)
\(\newcommand {\rO }{\operatorname {O}}\)
\(\newcommand {\rU }{\operatorname {U}}\)
\(\newcommand {\rSU }{\operatorname {SU}}\)
\(\newcommand {\ev }{\operatorname {ev}}\)
\(\newcommand {\bil }{\operatorname {Bil}}\)
\(\newcommand {\rad }{\operatorname {rad}}\)
\(\newcommand {\Span }[1]{\operatorname {span}\{#1\}}\)
\(\newcommand {\R }{\mathbb {R}}\)
\(\newcommand {\C }{\mathbb {C}}\)
\(\newcommand {\Z }{\mathbb {Z}}\)
\(\newcommand {\F }{\mathbb {F}}\)
\(\newcommand {\Q }{\mathbb {Q}}\)
\(\newcommand {\N }{\mathbb {N}}\)
\(\renewcommand {\P }{\mathbb {P}}\)
\(\newcommand {\I }{\mathrm {I}}\)
\(\newcommand {\half }{\tfrac 12}\)
\(\newcommand {\rel }{\mathrel {\mathrm {rel}}}\)
\(\renewcommand {\vec }[3]{(#1_{#2},\dots ,#1_{#3})}\)
\(\newcommand {\lst }[3]{#1_{#2},\dots ,#1_{#3}}\)
\(\newcommand {\plst }[3]{#1_{#2}+\dots +#1_{#3}}\)
\(\newcommand {\oplst }[3]{#1_{#2}\oplus \dots \oplus #1_{#3}}\)
\(\newcommand {\pplst }[3]{#1_{#2}\times \dots \times #1_{#3}}\)
\(\newcommand {\dlst }[3]{\lst {#1^{*}}{#2}{#3}}\)
\(\newcommand {\dlc }[4]{\lc #1{#2^{*}}#3#4}\)
\(\newcommand {\hmg }[3]{[#1_{#2},\dots ,#1_{#3}]}\)
\(\newcommand {\rng }[2]{#1,\dots ,#2}\)
\(\newcommand {\lc }[4]{#1_{#3}#2_{#3}+\dots +#1_{#4}#2_{#4}}\)
\(\newcommand {\plus }[2]{#1+\dots +#2}\)
\(\newcommand {\set }[1]{\{#1\}}\)
\(\newcommand {\abs }[1]{\lvert #1\rvert }\)
\(\newcommand {\ip }[1]{\langle #1\rangle }\)
\(\newcommand {\norm }[1]{\|#1\|}\)
\(\newcommand {\bx }{\mathbf {x}}\)
\(\newcommand {\be }{\mathbf {e}}\)
\(\newcommand {\bq }{\mathbf {q}}\)
\(\newcommand {\bu }{\mathbf {u}}\)
\(\newcommand {\by }{\mathbf {y}}\)
\(\newcommand {\bv }{\mathbf {v}}\)
\(\newcommand {\E }{\mathbb {E}}\)
\(\newcommand {\cI }{\mathcal {I}}\)
\(\newcommand {\cB }{\mathcal {B}}\)
\(\newcommand {\sub }{\subseteq }\)
\(\newcommand {\st }{\mathrel {|}}\)
\(\newcommand {\bw }[3]{#1\leq #2\leq #3}\)
\(\newcommand {\col }[3]{(#1_{#2})_{#2\in #3}}\)
\(\newcommand {\supp }{\mathrm {supp}}\)
\(\newcommand {\restr }[1]{_{|#1}}\)
\(\newcommand {\re }{\operatorname {Re}}\)
MA22020: Exercise sheet 1
Warmup questions
-
1. Let \(U\) be a subset of a vector space \(V\). Show that \(U\) is a linear subspace of \(V\) if and only if \(U\) satisfies the following conditions:
-
2. Which of the following subsets of \(\R ^3\) are linear subspaces? In each case, briefly justify your answer.
(a) \(U_1:=\set {(x_1,x_2,x_3)\st x_1^2+x_2^2+x_3^2=1}\) (b) \(U_2:=\set {(x_1,x_2,x_3)\st x_1=x_2}\) (c) \(U_3:=\set {(x_1,x_2,x_3)\st x_1+2x_2+3x_3=0}\)
-
3. Which of the following maps \(f:\R ^2\to \R ^2\) are linear? In each case, briefly justify your answer.
(a) \(f(x,y)=(5x+y,3x-2y)\) (b) \(f(x,y)=(5x+2,7y)\) (c) \(f(x,y)=(\cos y,\sin x)\) (d) \(f(x,y)=(3y^{2},x^3)\).
-
4. Let \(U,W\leq V\) be subspaces of a vector space \(V\).
When is \(U\cup W\) also a subspace of \(V\)?
Homework
-
5. Which of the following subsets of \(\C ^3\) are linear subspaces over \(\C \)? In each case, briefly justify your answer.
(a) \(U_1:=\set {(z_1,z_2,z_3)\st z_1z_2=1}\) (b) \(U_2:=\set {(z_1,z_2,z_3)\st z_1=\bar {z}_2}\) (c) \(U_3:=\set {(z_1,z_2,z_3)\st z_1+\sqrt {-1}z_2+3z_3=0}\)
-
6. Let \(V,W\) be vector spaces, \(\lst {v}1n\) a basis of \(V\) and \(\lst {w}1n\) a list of vectors in \(W\). Let \(\phi :V\to W\) be the unique linear map with
\(\seteqnumber{0}{}{0}\)
\begin{equation*}
\phi (v_i)=w_i,
\end{equation*}
for all \(1\leq i\leq n\). Show:
Deduce that \(\phi \) is an isomorphism if and only if \(\lst {w}1n\) is a basis for \(W\).
Please hand in at 4W level 1 by NOON on Thursday 16th October 2025