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{\bf A few review problems about subspaces and bases}
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These are some practice problems to help you get ready for the makeup quiz on subspaces and bases. You should review chapter 1, especially section 1.6.\\[2ex]
{\bf (0)} You should know the definitions of the various concepts: subspace, basis, linear combination, span, null space, etc.\\
{\bf A.} There are some problems in section 3.2 about subspaces/bases; where they say `the vector space V', you should interpret it as saying $\RR^n$. I suggest 3.2: 9, 10, 27, 29, 30, 31, 32 (In problems 31 and 32, take $V$ to be a subspace of $\RR^n$. The verb `generate' is synonymous with `span'.) \\
{\bf B. } Suppose that $\{\vv_1,\vv_2,\ldots,\vv_k\}$ is a basis for $W$, and that $\ww$ is a vector of $W$ that is different from $\vv_1,\vv_2,\ldots,\vv_k$. Show that $\{\ww, \vv_1,\vv_2,\ldots,\vv_k\}$ is {\em not} a basis for $W$.\\
{\bf C.} Suppose that $B = \{\vv_1,\vv_2,\vv_3,\vv_4\}$ is a basis for $W$. Which of the following sets of vectors are a basis for $W$?
\begin{enumerate}
\item $B_1 = \{ \vv_1 +\vv_2, \vv_2 +\vv_3, \vv_3 +\vv_4, \vv_4 +\vv_1\}$.
\item $B_2 = \{ \vv_1 -\vv_2, \vv_2 -\vv_3, \vv_3 -\vv_4, \vv_4 -\vv_1\}$.
\item $B_3 =\{17 \vv_1 +43 \vv_2- 22\vv_3 +6\vv_4, 19\vv_1 -13\vv_3 +\vv_4, 23 \vv_4 -184\vv_1\}$.
\end{enumerate}
{\bf D.} Let $A$ be an $m\times n$ matrix, and let $V$ be a subspace of $\RR^m$.
\begin{enumerate}
\item Let $W = \{ \bb \in \RR^m \ | \, \bb = A\vv\ \text{for some }\ \vv\in V\}$. Show, from the definition of subspace, that $W$ is a subspace.
\item If $\{\vv_1,\vv_2,\ldots,\vv_k\}$ is a basis for $V$, then show that $\{A\vv_1,A\vv_2,\ldots,A\vv_k\}$ spans $W$.
\item Is $\{A\vv_1,A\vv_2,\ldots,A\vv_k\}$ a basis for $W$? Give examples where it is a basis, and examples where it is not a basis. What property or properties of $A$ might ensure that it is a basis?
\end{enumerate}
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