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\begin{center}
{\Large Review for Test 2}
\end{center}

\begin{itemize}
\item  Test: Monday November 25.
\item  Test in Ritter 142 (downstairs!!)
\item  Test will cover 4.2 - 6.2, excluding the following topics 
  (everything else is ``fair game''):
  \begin{enumerate}
  \item  Fourier Series (projections in function spaces, section 4.3)
  \item  Least Squares (section 4.5)
  \item  Cramer's rule, but you do need to know what the adjoint matrix is and 
    how it is related to inverse matrices.
  \item  Matrix exponentials (nice application, but I won't test you on it).
  \end{enumerate}

\end{itemize}


\begin{enumerate}
\item  Definitions
  \begin{enumerate}
  \item  Orthogonal projection
  \item  Orthogonal basis, orthonormal basis.
  \item  Othogonal matrix.
  \item  Determinant, cofactor, minor, adjoint
  \item  Eigenvalue, eigenvector, eigenspace
  \item  Characteristic polynomial, characteristic equation.
  \item  Deficient
  \item  Diagonalizable, similar matrices
  \end{enumerate}

\item  Computations
  \begin{enumerate}
  \item  Gram-Schmidt: given a basis for a vector space, can you turn it into an orthogonal basis?
    How about an orthonormal basis?
  \item  How do you tell if a given matrix is orthogonal?  (Theorem~1 on page~249).
  \item  Computing determinant by expanding along a row or column (section~5.1).
  \item  Computing determinant by using row and column operations (section~5.2).
  \item  Be able to compute the adjoint of a matrix (section~5.3 and lecture).
    How does this give you the inverse of a matrix?
  \item  Finding eigenvalues (characteristic equation and characteristic polynomial).
  \item  Finding a basis for each eigenspace.
  \item  Determining if a matrix is deficient.
  \item  Diagonalizing a matrix (and using this to compute powers of a matrix).
  \end{enumerate}


\item  Theorems (how do these help us in our computations?)
  \begin{enumerate}
  \item  Theorem 1, page 220: This theorem tells us that if we want to find an 
    orthogonal basis, then the Gram-Schmidt process will alway work.
  \item  (Don't worry about the theorems on page~222 and page~224 for the test.)
  \item  Orthogonal matrices preserve angles (page~247).
    This tells us that orthogonal matrices preserve the geometry of the space.
    This will be more important when we get to section~6.5.
  \item  What is an orthogonal matrix theorem (page~249).
  \item  Computing determinant theorem (page~272): this basically says that you are 
    free to expand about any row or column.
  \item  Determinant of a transposed matrix theorem (page~272): this says that the determinant 
    of a matrix and its transpose are the same.
  \item  Determinant using row operations theorems (page~274, 275, 276, 281): 
    these theorems tell us how to use row operations to compute determinants.
  \item  Determinant nonzero means matrix is nonsingular theorem (page~283).
    Very important theorem - this gives us a way to tell if a matrix is singular or nonsingular
    (or not invertible or invertible).
  \item  (Don't worry about the uniqueness theorem on page~284 for the test)
  \item  Product theorem for determinants (page~284):
    using this we can deduce properties of determinants.
    For example, in class we used this to prove that the determinant of an orthogonal matrix is 
    always either $1$ or $-1$.
  \item  Cramer's rule: the important thing here is theorem~1 on page~292.
    From class, what this is really saying is something about the adjoint matrix and the inverse of $A$.
    \[
    A (\mathrm{adj}(A)) = (\det A) I
    \]
  \item  Eigenvalue and eigenvector theorems (when can you diagonalize a matrix?):  
    these theorems (mostly in section 6.2) tell conditions for being able to find enough
    eigenvectors to diagonalize a matrix.
    (In any case the proceedure to diagonalize a matrix is alway the same, as long as it is possible.)
  \end{enumerate}


\item  Sample problems (usual disclaimer: the test may or may not be like these problems)
  \begin{enumerate}
  \item  Consider the subspace of $\R^3$ with basis $\{ [1,2,3]^t, [2,-1,3] \}$.
    Find an orthogonal basis for this subspace.
    Find an orthonormal basis.
    (I wasn't careful about making vectors that work out nicely, but for a 2-dimensional subspace,
    it shouldn't get too nasty.)
  \item  Is the following matrix orthogonal?  (put your favorite matrix below)
    (Something similar to problems 4 and 10 of section 4.4 are similar and good to think about.)
  \item  Compute the determinant of the following matrix (put your favorite matrix below).
    Do this both using cofactor expansion (expanding along a row or column) and using row operations.
  \item  Compute the determinant and adjoint of the matrix below (put your favorite matrix below).
    Using the adjoint (assuming the determinant is nonzero) find the inverse of the matrix.
  \item  Find all eigenvalues and a basis for each eigenspace for the matrix below 
    (put your favorite matrix below).
    If the matrix is diagonalizable, diagonalize the matrix.
    If the matrix is not diagonalizable, explain why not.

  \end{enumerate}



\end{enumerate}

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