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\( \newcommand {\multicolumn }[3]{#3}\)
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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?
-
2. Let \(V_i\leq V\), for \(\bw 1ik\). Prove the converse of Corollary 2.8: if
\(\seteqnumber{0}{}{0}\)
\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:
\(\seteqnumber{0}{}{0}\)
\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:
\(\seteqnumber{0}{}{0}\)
\begin{equation*}
U/(U\cap W)\cong (U+W)/W.
\end{equation*}
-
(b) Deduce that, when \(V\) is finite-dimensional,
\(\seteqnumber{0}{}{0}\)
\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 31st October 2024