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{\Huge 189-251B: Algebra II} \\ \sk
{\Huge Assignment 1} \\ \sk
{\Large Due: Wednesday, January 15}
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\noindent
1. Show that the set $V=\R^2$, with the addition and scalar multiplication
defined by the rules
$$ (x_1,y_1) + (x_2,y_2) := (x_1+x_2, y_1+y_2), \quad
\lambda \cdot (x,y) := (\lambda x,0) $$
satisfies all the axioms of a vector space {\em except } the property that
$ 1\cdot v = v$ for all $v\in V$.
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2. Which of the following subsets of $\R^3$ are
$\R$-vector subspaces of $\R^3$?
\begin{enumerate}
\item The set of vectors $(x,y,z)$ with $x\ge 0$.
\item The set of vectors $(x,y,z)$ satisfying $x+y=2z$.
\item The set of vectors satisfying $x=0$ or $y=0$.
\item The set of vectors satisfying $x=0$ and $y=0$.
\item The set of vectors with rational coordinates $x,y,z$.
\end{enumerate}
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3. Let $V_1$ and $V_2$ be two vector subspaces of an $F$-vector space $V$.
Show that the union
$V_1\cup V_2$ can only be a subspace of $V$ if one of
the spaces $V_i$ ($i=1$ or $2$) is contained in the other.
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4. Let $V$ be the vector space of $n\times m$ matrices with
entries in a field $F$. What is the dimension of $V$?
Give an explicit basis for $V$ over $F$.
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5. Let $v_1$, $v_2$, and $v_3$ be three linearly independent vectors in an
$\R$-vector space $V$. Show that the vectors
$v_1+v_2$, $v_2+v_3$, and $v_3+v_1$ are also linearly independent.
What if the field $\R$ is replaced by the field $\Z_2$ in this question?
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6. Let $V$ be the $\R$-vector space of all
infinitely differentiable functions on the real line.
Show that the function $T:V\rightarrow V$ defined by $T(f) = f'$
(where $f'$ denotes as usual the derivative of $f$) is a linear
transformation from $V$ to itself.
Show that $T$ is not injective, and compute its kernel.
Show that $T$ is surjective. (Hint: use the fundamental theorem of calculus!)
Conclude that $V$ is not finite dimensional.
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7. A {\em generalised vector space} over a field $F$ is a {\em not necessarily
commutative} group $V$ (so that the group operation is written
using the multiplicative notation) equipped with a ``scalar multiplication"
$$ F\times V \rightarrow V \quad \mbox{ denoted }\quad
(\lambda,v) \mapsto v^{[\lambda]},$$
satisfying the following axioms analogous to those of a usual vector space
\begin{itemize}
\item[M1] $(vw)^{[\lambda]} = v^{[\lambda]} w^{[\lambda]}$ for all $v,w\in V$ and $\lambda\in F$;
\item[M2] $v^{[\lambda_1+\lambda_2]} = v^{[\lambda_1]} v^{[\lambda_2]}$, for all
$v\in V$ and $\lambda_1,\lambda_2\in F$;
\item[M3] $ (v^{[\lambda_1]})^{[\lambda_2]} = v^{[\lambda_1\lambda_2]}$, for all
$v\in V$, and $\lambda_1,\lambda_2\in F$;
\item[M4] $v^{[1]}=v$, for all $v\in V$.
\end{itemize}
Show that a generalised vector space is just an ordinary vector space: i.e., the group law on $V$ is necessarily commutative.
(Hint: Show that $v^{[-1]} = v^{-1}$, the latter expression being
the inverse of $v$ in the group $V$, and consider axiom $M1$
with $\lambda=-1\in F$.)
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8. Let $X$ be a set, and let $\cP(X)$ denote the {\em power set} of $X$, i.e.,
the set of all subsets of $X$. Define the sum of two sets to be
$$ A + B := A\cup B - (A\cap B),$$
and define a scalar multiplication of $\Z_2$ on $\cP(X)$ by the rule:
$$ 0\cdot A := \emptyset, \quad 1\cdot A = A.$$
Show that $\cP(X)$ with these operations is a vector space over $\Z_2$.
What is its dimension?
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