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\begin{flushright}
  Your Name Goes Here
\end{flushright}


\begin{center}
  \textbf{Math 24}\\ Homework 4
\end{center}



\begin{itemize}


\item [\#3.1.9] Suppose that $T\in \L(V,W)$ is injective and that
  $\{v_1, \dots, v_n\}$ is linearly independent in~$V$.  Show that
  $\{T(v_1), \dots, T(v_n)\}$ is linearly independent in $W$.

\item [\#3.1.12] Suppose that $A \in M_n(\F)$ is upper triangular (see
  Exercise 2.3.12) and that all the diagonal entries are nonzero
  (i.e., $a_{ii} \ne 0$ for all $i=1, \dots, n$.)  Use Corollary 3.7
  to prove that the columns of $A$ are linearly independent.
 

\item [\#3.1.14] Let $n \ge 1$ be an integer, and suppose that there
  are constants $a_1, \dots, a_n\in \R$ such that
  \[\sum_{k=1}^n a_k \sin(kx) = 0 \]
  for every $x \in \R$.  Show that $a_1=\cdots=a_n=0$.\newline
  \textit{Hint:} Consider the linear map
  $D^2:C^\infty(\R) \to C^\infty(\R)$ given by $D^2(f) = f''$, where
  $f''$ is the second derivative and $C^\infty(\R)$ is the space of
  infinitely differentiable functions on $\R$, and use Theorem 3.8.
  
\item [\#3.2.10] Find the unique matrix $A \in M_2(\R)$ such that
  $ A\begin{bmatrix} 1\\2
  \end{bmatrix}=
  \begin{bmatrix*}[r]
    0\\-1
  \end{bmatrix*}$ and
  $A\begin{bmatrix} 2\\1
  \end{bmatrix}=
  \begin{bmatrix*}[r]
    -3\\2
  \end{bmatrix*}$.

\item [\#3.2.16] Find a sublist of $\{x+1, x^2-1,x^2+2x+1,x^2-x\}$
  which is a basis for $\P_2(\R)$.

\item [\#3.2.22] Suppose that $\{v_1, \dots, v_n\}$ and $\{w_1, \dots,
  w_n\}$ are both bases for $V$.  Show that there is an isomorphism
  $T\in \L(V)$ so that $T(v_i) = w_i$ for all $i$.
  
\end{itemize}

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