\magnification=\magstep1
\centerline{Mathematics 244, section 1 -- Linear Algebra}
\centerline{Practice True-False Questions for Exam 2}
\centerline{March 19, 2004}
\bigskip
\noindent
{\it For each true statement, give a short proof or example; for each false
statement give a reason or a counterexample}
\bigskip
\item{A)}  If $T : V \to W$ is linear, $\dim(V) = 5$, $\dim(W) = 7$
then $T$ is surjective.
\bigskip
\item{B)}  Every linear $T : V \to W$, where $\dim(V) = 5$, $\dim(W) = 7$,
is injective.
\bigskip
\item{C)}  Some linear $T : V \to W$, where $\dim(V) = 5$, $\dim(W) = 7$,
is injective.
\bigskip
\item{D)}  If $\{v_1, v_2, \ldots, v_n\}$ in $V$ is linearly dependent, 
and $T : V \to W$ is linear, then 
$$\{T(v_1),\ldots, T(v_n)\}$$ 
in 
$W$ is linearly dependent.
\bigskip
\item{E)}  If $T: {\bf R}^n \to {\bf R}^n$ is linear, then 
$Ker(T) \cap Im(T) = \{\vec{0}\}$.
\bigskip
\item{F)}  If $T : V \to V$ is linear, and $\beta = \{v_1,\ldots,v_n\}$,
$B' = \{cv_1,\ldots,cv_n\}$, $c \ne 0 \in {\bf R}$ are bases for V, then
the matrices $A,A'$ of $T$ with respect to $\beta,\beta$ and $\beta',\beta'$ are equal.
\bigskip
\item{G)}  If ${\bf A}$ is an $n\times n$ matrix, and $c$ is a fixed real
number, the set
$W =\{v \in {\bf R}^n : Av = cv\}$ is a vector subspace of ${\bf R}^n$.
\bigskip
\item{H)}  If $T = I_{{\bf R}^2}$ is the identity mapping and $\beta$, $\beta'$
are two bases for ${\bf R}^2$, then the matrix of $T$ with respect
to $\beta,\beta'$ is $\pmatrix{1&0\cr 0&1\cr}$.
\bigskip
\item{I)}  If $T : V \to W$ is an isomorphism, and $U \subseteq V$ is a
vector subspace, then $\dim(U) = \dim(T(U))$.
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\item{J)}  If $v_1,\ldots, v_n$ are vectors in a vector space $V$, and 
$w_1,\ldots, w_n$ are vectors in a vector space $W$, then there
always exists some linear $T : V \to W$ satisfying $T(v_i) = w_i$ for 
all $i = 1,\ldots, n$.
\bigskip
\item{K)}  If $T: {\bf R}^n \to {\bf R}^n$ is linear, and
$Ker(T) + Im(T) = {\bf R}^n$, then $Ker(T) \cap Im(T)=\{\vec{0}\}$.
\bye

