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%Example of use of oxmathproblems latex class for problem sheets
\documentclass{oxmathproblems}
%(un)comment this line to enable/disable output of any solutions in the file
%\printanswers
%define the page header/title info
\oxfordterm{MT18}
\course{Impossible Maths I}
\sheetnumber{3}
\sheettitle{First topic questions} %can leave out if no title per sheet
% add further contact details to footer if desired,
%e.g. email address, or name and email address
\contact{Joe Bloggs: joe.bloggs@maths.ox.ac.uk}
\begin{document}
\begin{questions}
\miquestion
\begin{parts}
\part Define what it means for a finite subset of a vector space to be
{\em linearly independent}, a {\em spanning set}, and a {\em basis}.
State the {\it Steinitz Exchange Lemma}.
Prove that if a vector space $V$ has a finite basis, then every linearly
independent subset of $V$ may be extended to a basis.
Prove that any two finite bases of a vector space have the same number of
elements. Define the {\em dimension} of a finite-dimensional vector space.
\part Let $V$ be a finite-dimensional vector space.
Suppose that $X$ and $Y$ are subspaces of $V$.
Prove that
\[ \dim(X+Y)+\dim(X\cap Y)=\dim X+\dim Y. \]
\part Suppose $V$ is a finite-dimensional vector space, and $T:V\to V $
is a linear transformation. Suppose that for all $v\in V$, if $T(T(v))=0$,
then $T(v)=0$. Prove that $V=\ker T\oplus \mathop{\rm im} T$, and that the
restriction, $T{\restriction} {\mathop{\rm im} T}$, of $T$ to $\mathop{\rm
im} T$ is a bijection from $\mathop{\rm im} T$ to itself.
[{\em You may assume the Rank-Nullity Theorem.}]
[{\em If\/ $V$ and $W$ are vector spaces, $T:V\to W$ is a linear
transformation,
and $U$ is a subspace of $V$, then we define the {\em restriction}
$T{\restriction} U$ of $T$ to $U$ to be a linear transformation from $U$
to $W$ such that for all $u\in U$, $(T{\restriction} U)(u)=T(u)$.}]
\end{parts}
\begin{solution}
The solution would go here
\end{solution}
\miquestion
\begin{parts}
\part Define an {\em elementary row operation}, and say what it means
for a matrix to be in {\em row-reduced echelon form}. Describe how any
matrix may be reduced to row-reduced echelon form.
\part For which values of $\lambda$ is the following system of equat
ions solvable? In each case calculate the solutions.
\[
\centerline{\hbox{\vbox{\openup1.5\jot\halign{\hss$\displaystyle{}\ #\
{}$\hss&& \hss$\displaystyle {}\ #\ {}$\hss\cr x&+2y&-3z&=&5\cr
x&-2y&-5z&=&7\cr 2x&+8y&+(-\lambda-6)z&=&8\cr
x&-2y&+(\lambda-3)z&=&\lambda^2+3\cr }}}}
\]
\end{parts}
\begin{solution}
The solution would go here
\end{solution}
%force a page break for better layout of questions
%NOTE: only force pagebreaks at the final stage for perfecting the layout
%\newpage
\miquestion
\begin{parts}
\part
\begin{subparts}
\subpart Define what it means to say that a square matrix with real
entries is {\em diagonalisable} over ${\mathbb R}$. \subpart Show that
if $A$ is a square matrix with real entries and $\bf u$ and $\bf v$ are
eigenvalues of $A$ corresponding to different eigenvalues, then $\bf u$
and $\bf v$ are linearly independent.
\end{subparts}
\part
\begin{subparts}
\subpart Consider the $2\times 2$ real square matrix
\[ A=\begin{pmatrix}a&b\cr b&d\cr\end{pmatrix}. \] Show that $A$ has at
least one real eigenvalue; and that if $a\neq d$ or $b\neq 0$, then it
has two distinct real eigenvalues.
Deduce that $A$ is diagonalisable.
\subpart Determine the values of $\alpha$ and $\beta$ for which the real
matrix
\[ A=\begin{pmatrix}1&\alpha\cr \beta&1\cr\end{pmatrix} \] is
diagonalisable.
\subpart Determine when the $2\times 2$ real matrix
\[ A=\begin{pmatrix}a&b\cr c&a\cr\end{pmatrix} \] is diagonalisable.
\end{subparts}
\end{parts}
\begin{solution}
The solution would go here
\end{solution}
\miquestion Let $\mathbf{u}$, $\mathbf{v}$ and $\mathbf{w}$ be vectors in
$\mathbb{R}^3$.
\begin{parts}
\part[7]
\begin{subparts}
\subpart Prove that $\mathbf{u}$, $\mathbf{v}$ and $\mathbf{w}$ are linearly
independent if and only if
$\mathbf{u}.(\mathbf{v}\mathbin{\land}\mathbf{w})\neq \mathbf{0}$.
\subpart Establish the identity
\[
\mathbf{u}\mathbin{\land}(\mathbf{w}\mathbin{\land}\mathbf{v})
=(\mathbf{u}.\mathbf{v})\mathbf{w}-(\mathbf{u}.\mathbf{w})\mathbf{v}.
\]
\end{subparts}
\part[7] Prove that $\mathbf{u}$, $\mathbf{v}$ and $\mathbf{w}$ are linearly
independent if and only if
$\mathbf{v}\mathbin{\land}\mathbf{w}$,
$\mathbf{w}\mathbin{\land}\mathbf{u}$, and
$\mathbf{u}\mathbin{\land}\mathbf{v}$ are linearly independent.
\part[6] Suppose that $\mathbf{u}$, $\mathbf{v}$ and $\mathbf{w}$ are
linearly independent, and that
\[ \mathbf{r}=a\mathbf{u}+b\mathbf{v}+c\mathbf{w}. \]
Find coefficients $\alpha$, $\beta$ and $\gamma$ such that
\[
\mathbf{r}=\alpha\mathbf{v}\mathbin{\land}\mathbf{w}
+\beta\mathbf{w}\mathbin{\land}\mathbf{u}
+\gamma\mathbf{u}\mathbin{\land}\mathbf{v}.
\]
\end{parts}
\begin{solution}
The solution would go here
\end{solution}
\end{questions}
\end{document}
```