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{\bf 21-341} \hfill {\bf Linear Algebra I} \hfill {\bf Fall 2003}\\

\centerline{\bf Review Problems for Test 1} 

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\begin{enumerate} 
\item 
Let $\mathbb{F} = \mathbb{R}$ and $V = \mathbb{R}^4$.  Determine whether or not the list of vectors is linearly independent. 

\begin{enumerate} 
\item $<1,1,2,1>, \ \ \ <1,1,2,2>$ 

\item $<1, -1,1,0>,\ \  <1,2,3,4>,  \ \  <1,2,3,1>,<-1,1,-1,1>,\\  <1,0,1,0>$

\item $<1,-1,1,1>, \ \ <0,0,0,0>, <1,3,2,1>$ 

\item $<1,2,-1,0>, \ \ <2,1,0,1>,\ <4,5,-2,4>$
\end{enumerate}

\item Let $\mathbb{F} = \mathbb{R}$ and $V = \mathbb{R}^5$.  Find a basis for the solution set of the equation $x_1+2x_2 - x_3+x_4-2x_5=0$. 

\item Prove or Disprove:  Let $V$ be a finitely generated vector space and let $S,T,R$ be subspaces of $V$.  Then $S+(T\cap R)=(S+T)\cap(S+R)$. 

\item Let $\mathbb{F} = \mathbb{R}$ and let $n \in \mathbb{Z}^+$ be given.  Let $P_n(\mathbb{R})$ denote the vector space of all real polynomials of the degree $\leq n$.  Assume that $f_0,f_1, \ldots f_n \in P_n (\mathbb{R})$ satisfy $f_0(\pi)=f_1(\pi) = f_2(\pi) = \ldots = f_n(\pi) = 0$. Show that the list $f_0,f_1,f_2, \ldots, f_n$ is linearly dependent. 

\item Let $\mathbb{F}$ be a field and $V,W$ be vector spaces over $\mathbb{F}$.  Let $L:V \rightarrow W$ be a mapping such that $L(u +v)=L(u)+L(v)$ and $L(\lambda u) = \lambda L(u)$ for all $u, v \in V,\ \lambda \in \mathbb{F}$.  Let

$$T = \{L(u): u \in V\}.$$ 

Show that $T$ is a subspace of $W$.  

\item Let $\mathbb{F} = \mathbb{R}$ and let $P_3(\mathbb{R})$ be the vector space of all real polynomials of degree $\leq 3$.  Let $S =\{f \in \mathbb{P}_3(\mathbb{R}): f(1) = 2 {\displaystyle \int^1_0}f(x)dx\}.$

Show that $S$ is a subspace of $P_3(\mathbb{R})$ and find a basis for $S$. 

\item Let $\mathbb{F} = \mathbb{R}$ and $A = \left(\begin{array}{llllll} 
1& 2& 1& -1\\
1& 5& 2& 1\\
-2& 5& 5 & 4\end{array} \right)$.

Find a $3 \times 2$ matrix $A'$ such that $A'$ is row equivalent to $A$, all zero rows of $A'$ are below the nonzero rows, and the nonzero rows of $A'$ are in echelon form. 

\item Let $V$ be an eight dimensional vector space and let $S,T$ be subspaces of $V$ with ${\rm dim}(S)=5$ and ${\rm dim}(T)=6$.  What is the smallest possible dimension of $S \cap T$.

\item Prove or Disprove:  Let $S,T,R$ be subspaces of a finitely generated vector space $V$.  If $S+T = S+R$ then $T=R$.   

\item In this problem, we will write complex numbers in the form $a+bi$ where $a,b \in \mathbb{R}$ and $i^2=-1$; for such a number we write 

$$Re(a+bi)=a, \ \ \ \ \ Im(a+bi)=b. $$ 

Let $V = \mathbb{C}^4$. 

\begin{enumerate} 

\item Assume that $\mathbb{F}=\mathbb{R}$.  Which of the following are subspaces? 

\begin{enumerate} 

\item $\{<z_1,z_2,z_3,z_4>:\ z_1+z_4=0\}$

\item $\{<z_1,z_2,z_3,z_4>:\ z_1+z_4=1\}$ 

\item $\{< z_1,z_2,z_3,z_4>:\ z_1+z_4=z_2\}$ 

\item $\{< z_1,z_2,z_3,z_4:\ Re(z_1)=0\}$

\item $\{< z_1,x_2,z_3,z_4>:\ Im(z_1)=0\}$ 

\item $\{< z_1,z_2,z_3,z_4>:\ z_1z_2=0\}$

\end{enumerate} 

\item How would the answers to part (a) change if the field were changed from $\mathbb{R}$ to $\mathbb{C}$. 

\end{enumerate} 

\item Let $\mathbb{F}=\mathbb{R}$ and $V={\cal F}(\mathbb{R})$ the set of all real-valued functions $f: \mathbb{R} \rightarrow \mathbb{R}$.  Let $f_1(x)=e^x, \ \ f_2(x)=e^{2x},\ \ f_3(x) = e^{3x}$. Determine whether or not $f_1,f_2,f_3$ are linearly independent.  


\end{enumerate} 







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