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\title{Math 23a Problem Set 3}
\dedicatory{due October 13th}

\begin{document}
\maketitle

\begin{enumerate}
 \item Using the axioms of Vector spaces as in your book [p.~189], prove that
$0v=\vec 0\forall v\in V$.

 \item Let $F\colon V\to V'$ be a linear map, $L=\ker F, W=\image F$. Let
$l_1,\dots,l_r$ be vectors in $L$ and $v_1,\dots,v_m$ be vectors in $V$. For all
$i, 1\leq i\leq m$ we define $w_i=F(v_i)\in W$.

\begin{enumerate}
\item Assume that the vectors $\set{l_1,\dots,l_r}\in L$ are linearly independent and
that the vectors $\set{w_1,\dots,w_m} \in W$ are  linearly independent. Prove
that the vectors $\set{l_1,\dots,l_r,v_1,\dots,v_m} \in V$ are also linearly
independent. 

\item Assume that the vectors $\set{l_1,\dots,l_r}$ span $L$  and
that the vectors $\set{w_1,\dots,w_m}$ span $W$.  Prove
that the vectors $\{l_1,\dots,l_r,$
$v_1,\dots,v_m \}$ span $V$.

\item Assume that the vectors $\set{l_1,\dots,l_r}$ are a basis of $L$   and
that the vectors $w_1,\dots,w_m $ are a basis of $W$. Prove
that the vectors $\{l_1,\dots,l_r,v_1,\dots,v_m \}$ are a basis of $V$.
\end{enumerate}

 \item Let $V$ be a two-dimensional vector space, and $\set{e_1,e_2}$
and $\set{v_1,v_2}$ be two bases of $V$, such that $v_1=e_1
+e_2,v_2=e_1 -e_2$.
Let $e^1 ,e^2$ and $v^1,v^2$ be the corresponding dual bases of $V^*$.
Find the expression of $v^1,v^2$ in terms of  $e^1 ,e^2$.

 \item Let $F\colon\bR^n \to \bR$ be the map given by 
$$\begin{bmatrix}x_1\\ x_2\\
\vdots\\x_n\end{bmatrix} \mapsto \sum_{i=1}^n x_i.$$
Find the dimension of $\ker F$.

 \item Hubbard, 2.6.7
\end{enumerate}
\end{document}
