7-2.tex

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\begin{document}
\begin{center}
    7/2
\end{center}
\begin{enumerate}
    \item
        Prove that $T$ is one-to-one if and only if $T(x)=0$ implies $x=0$.
    \item
        Let $T:V\to W$ be a linear map. Let $S$ be a linearly independent
        subset of $V$. Prove that if $T$ is one-to-one, then $T(S)$ is linearly
        independent. Give a counterexample when $T$ is not one-to-one.
    \item
        Let $T:V\to W$ be a linear map. Let $S$ be a spanning
        subset of $V$. Prove that if $T$ is onto, then $T(S)$ is 
        spanning. Give a counterexample when $T$ is not onto.
    \item
        Let $T:V\to W$ and $B = \{v_1,\ldots,v_n\}$ be a basis for $V$ and
        $\{w_1,\ldots,w_n\}\subseteq W$. Prove that there is exactly
        one linear transformation such that $T(v_i)=w_i$ for all
        $i=1,\ldots,n$. What could go wrong if $B$ is not spanning? What could
        go wrong if $B$ is not linearly independent.
    \item
        Let $T:V\to W$ and $U:V\to W$. Let $B$ be a basis for $V$, prove that
        if $T(b)=U(b)$ for all $b\in B$, $T=U$.
    \item
        Identity the polynomial $ax^2+bx+c$ with the vector $(a,b,c)$. What is
        the matrix corresponding to $\frac{d}{dx}:P_2(\mathbf{R})\to
        P_2(\mathbf{R})$. What is the kernel? What is the range? 
    \item
        What is the kernel of differentation from $C^\infty(\mathbf{R})\to
        C^\infty(\mathbf{R})$?
\end{enumerate}

    
\end{document}