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{\bf
\noindent
LINEAR ALGEBRA (Prof. De Loera) \hskip 4cm \hfill \null \\
Graduate Proficiency Exam  \hskip 5.4cm Fall 2005 \hfill \null \\}

\noindent These problems are intended to help you review material
covered in the prelim. Some problems are harder and longer than what
you will find in the exam. Try to write the solutions as you would
in the exam: Write all details when solving every problem. 
Be organized and use the notation appropiately. Initially try to solve 
the problems without any assistance.

\begin{enumerate}


\item Given the matrix
$A=\left[ \begin {array}{ccccc} 2&-3&-7&5&2\\\noalign{\medskip}1&-2&-4&3
&1\\\noalign{\medskip}2&0&-4&2&1\\\noalign{\medskip}1&-5&-7&6&2
\end {array} \right]$ find a basis for (a) $range(A)$, (b)
  $kernel(A)$.

Determine whether the vector $[2,2,-3,7]$ is in the range of $A$ or
not.

\item Let $A= \left[ \begin {array}{ccc}
 5&-6&-6\\\noalign{\medskip}-1&4&2 \\\noalign{\medskip}3&-6&-4\end
 {array} \right]$. Find the eigenvalues of $A$ and if possible find a
 matrix $S$ such that $S^{-1} A S$ is a diagonal matrix.

\item Can the matrix 

$\left[ \begin {array}{ccc} 0&0&1\\\noalign{\medskip}0&1&0
\\\noalign{\medskip}1&0&0\end {array} \right]$ be unitarily
  diagonalized? If yes, do it!

\item Given the matrix 
$
 \left[ \begin {array}{cccc} 0&1&0&0\\\noalign{\medskip}0&0&1&0
\\\noalign{\medskip}0&0&0&1\\\noalign{\medskip}0&6&-11&6\end {array}
 \right]
$
Find its eigenvalues, its characteristic polynomial and if possible 
diagonalize it (find explicit $S$ such that $S^{-1} A S$ is diagonal).

\item The linear transformation given by derivation $d/dt$ acts on the
vector space of all polynomials with degree at most 3. Represent this
linear transformation on the basis given by $\{1,t,t^2,t^3\}$. What is
the Jordan canonical form of this matrix?

\item (a) What are the possible Jordan forms of an $n \times n$ matrix $A$
such that $A^3=I$? 

(b) What are the possible Jordan forms of a matrix $A \in M_6$ with
characteristic polynomial $p_A(t)=(t+3)^4(t-2)^2$?


\item Apply the Gram-Schmidt procedure to the vectors $x_1=(1,1,0)$,
$x_2=(2,0,1)$ and $x_3=(2,2,1)$.

\item Prove or disprove: Let $A=\{x_1,x_2,\dots,x_n\}$ be
linearly independent vectors, then $A \cup \{y \}$ is a set of
linearly independent vectors if and only if $y \notin span(A)$.


\item A linear transformation $P$ is a projection if $P^2=P$. Prove
 
a) $P$ is a projection if and only if $I-P$ is a projection.

b) $y \in range(P)$ if and only if $Py=y$.

c) $range(P)=kernel(I-P)$.

d) the vector space $V$ is the direction sum of $range(P)$ and
$kernel(P)$.

\item (a) If $\Lambda$ denotes the diagonal $n \times n$ matrix with
  eigenvalues $\lambda_1,\dots,\lambda_n$. Show that $\Lambda B=
  B\Lambda$ for some $B$, $n \times n$ matrix, if and only if $B$ is
also diagonal.

(b) Suppose $A \in M_n$ has $n$ distinct eigenvalues. If $AB=BA$ for
  some $B \in M_n $, show that $A,B$ are simultaneously diagonalizable.

\item Let $A \in M_n$. If $rank(A)=1$, then there exists a scalar
  $\beta \in {\mathbb C}$ such that $A^2=\beta A$.

\item Given a $4 \times 4$ matrix of rank 3, explain which are
the possible $A$-invariate subspaces of ${\mathbb C}^4$.


\item Are there real $3 \times 3$ matrix whose minimal polynomial is
$x^2+1$? How about if the matrix is complex ? What happens if the
  matrix is $2 \times 2$?

\item True or false (justify):

a) Similar matrices have the same minimal and characteristic
polynomial

b) If two matrices have the same minimal and characteristic polynomial
they must be similar.

c) inverse of an upper triangular matrix is upper triangular.

d) Let $W_1,W_2$ be subspaces of $V$. Then $W_1 \cap W_2$ is a
subspace too.

e) Let $W_1,W_2$ be subspaces of $V$. Then $W_1 \cup W_2$ is a
subspace too.

f) Let $A$ be a $k \times n$ matrix. $dim(range(A))+dim(Kernel(A))=n$

g) Let $A \in M_{k,n}$ and $B \in M_{n,k}$. If $tr(X)$ denotes the
trace of $X$, then $tr(AB)=tr(BA)$.

\item Let $A,B \in M_n$. Prove or disprove that

a) $range(A+B) \subset range(A)+range(B)$.

b) $range(AB) \subset range(A)$.

c) $rank(AB) \leq rank(B)$.

d) If $A$ is non-singular then $rank(AB)=rank(B)=rank(BA)$.

\item Suppose $A,B \in M_n$ commute and have eigenvalues
  $a_1,\dots,a_n$ and $b_1,\dots,b_n$ respectively. Explain, with
a proof, what are the
eigenvalues of $AB$ 

\item 

a) State definitions of normal matrix, nilpotent matrix, and Hermitian matrix. 

b) Assume that $A$ is normal and that $A,B \in M_n$ commute. Prove that $A^*$ and $B$ commute too.

c)When is a normal matrix unitary? When is it Hermitian? When is it nilpotent?

d) State three conditions equivalent to normality.

\item Let $W,Y$ be subspaces of $V$ such that $V=W+Y$. If $W \cap
  Y=\{0\}$, every vector $v\in V$ has a unique representation in the
  form $v=x+y$ with $x \in W, \ y \in Y$.

\item Let $A \in M_n$ and $x,y$ represent non-zero vectors such that
$Ax=3x$ and $Ay=5y$. Prove that $x,y$ are linearly independent. How
general is this fact?

\item Prove or disprove: 

a) $rank(A^2) \leq rank(A)$.

b) $A,B \in M_n$ with $AB$ diagonalizable implies $BA$ diagonalizable.

c) If $P$ is a projection its eigenvalues are only $0,1$.

d) Let $P,Q$ be projections which are simultaneously diagonalizable,
then $PQ$ is a projection too.

\item Sally is playing ``matrix of fortune'' and knows that $4,-7,2$
are eigenvalues of the following matrix
$
 \left[ \begin {array}{cccc} 2&*&*&6\\\noalign{\medskip}*&3&*&*
\\\noalign{\medskip}*&*&-1&*\\\noalign{\medskip}*&*&*&5\end {array}
 \right]
$
To win the game she needs to find $det(A)$. Please help her!

\item Prove or disprove (with justification):

a) If $A,B$ are two $n \times n$ matrices and $A=A^t$ and $B=B^t$ then
$AB=(AB)^t$.

b) If $A$ is nilpotent matrix then $0$ is the only eigenvalue of $A$.

c) Similar matrices have the same eigenvalues.

d) If $\lambda$ is an eigenvalue of both $A,B$ then $\lambda$ is an
eigenvalue of $A+B$.

e) Matrices with the same determinant have the same rank.

f) If $\lambda$ is an eigenvalue of an orthogonal transformation then
$\lambda=\pm 1$

g) Matrices with the same characteristic polynomial are similar.

h) The set of all $n\times n$ real matrices with determinant zero is a
subspace of $M_n$.

i) If a square matrix $A$ is diagonalizable so is $A^2$.

j) Similar matrices have the same eigenvectors.

k) If $V,W$ are two distinct 2-dimensional subspaces of $R^3$, then
$V+W=R^3$.

l) Let $\{x,y,z\}$ be a linearly independent set of vectors in a
vector space over a finite field. The set $\{x+y,x+z,y+z\}$ is always
linearly independent.

\item Let $V$ and $W$ be the following subspaces of ${\mathbb R}^4$.

$V=span\{(1,3,-4,7),(-1,2,-1,3)\},\  W=span\{(2,4,3,1),(2,2,-1,3)\}$.

Determine bases for the subspaces $V+W$, $W\cap V$, and $V^{\perp}$.

\item Given the matrix $ A=\left[ \begin {array}{ccc} 21&6&-12\\\noalign{\medskip}6&12&-6
\\\noalign{\medskip}-12&-6&21\end {array} \right]$

Find the eigenvalues and an orthonormal basis for each of the
eigenspaces. Is this matrix diagonalizable over R? Can you find a matrix $B$,
such that $B^2=A.$?

\item  Determine whether the following matrices are diagonalizable
  over the indicated field. If possible find the diagonalization.

$ \left[ \begin {array}{ccc} 1&-1&2\\\noalign{\medskip}1&-1&2
\\\noalign{\medskip}1&-1&2\end {array} \right] \ \hbox{over the
    real numbers}$

$ \left[ \begin {array}{ccc} 3&0&1\\\noalign{\medskip}-1&2&-1
\\\noalign{\medskip}1&2&5\end {array} \right] \ \hbox{over the real numbers}$

$ \left[ \begin {array}{cc} 3&-4\\\noalign{\medskip}4&3\end {array}
 \right] \ \hbox{over the complex numbers}.$

\item Let $A$ be a normal and idempotent matrix. Then $\sum_{i,j} |a_{ij}|^2=rank(A)$.

\item Let $A= \left[ \begin {array}{cccc} 0&1&0&1\\\noalign{\medskip}1&0&1&0
\\\noalign{\medskip}0&1&0&1\\\noalign{\medskip}1&0&1&0\end {array}
 \right].$ Find the Jordan canonical form of $A$.


\item Let $A= \left[ \begin {array}{ccc} 3&-3&6\\\noalign{\medskip}-3&2&-9
\\\noalign{\medskip}6&-9&2\end {array} \right].$

Is this matrix diagonalizable? If yes, please find the
diagonalization. Show that $A$ determines an indefinite quadratic
form. Determine vectors $x,y$ such that $x^t A x >0$ and $y^tAy<0$.
How many eigenvalues are negative, positive, or zero?

\item  For an $m
  \times n$ matrix $A$:

Prove that the following conditions are equivalent
a) $rank(A)=n$, b) $A^tA$ is invertible, c) $BA=I$ for some $n\times
m$ matrix $B$.

Prove that the eigenvalues of $A^tA$ are non-negative and the same
as those of $AA^t$.

\item Are the eigenspaces of a matrix always orthogonal to each other?


\end{enumerate}

\end{document}







