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Exam\InsetSpace ~
3 
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Math\InsetSpace ~
314
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Fall 2006
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Name:
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Score:
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Instructions:
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 Show your work in the spaces provided below for full credit.
 Use the reverse side for additional space, 
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but clearly so indicate.
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 You must clearly identify answers and show supporting work to receive any
 credit.
 Exact answers (e.g., 
\begin_inset Formula $\pi$
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) are preferred to inexact (e.g., 
\begin_inset Formula $3.14$
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).
 Point values of problems are given in parentheses.
 Notes or text in 
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 Calculator is allowed.
 
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(25) 
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1.

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 Let 
\begin_inset Formula $\mathbf{v}_{1}=\left(1,1,0\right))$
\end_inset

, 
\begin_inset Formula $\mathbf{v}_{2}=\left(-1,1,1\right)$
\end_inset

, 
\begin_inset Formula $\mathbf{v_{3}}=\left(1/2,-1/2,1\right)$
\end_inset

 and 
\begin_inset Formula $\mathbf{v}=\left(1,2,-2\right)$
\end_inset

.
 
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\begin_layout Standard
(a) Find the norm of 
\begin_inset Formula $\mathbf{v}$
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.
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(b) Find the cosine of the angle between the vectors 
\begin_inset Formula $\mathbf{v}$
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  and 
\begin_inset Formula $\mathbf{v}_{1}.$
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(c) Verify the CBS inequality for the pair of vectors 
\begin_inset Formula $\mathbf{v},\mathbf{v}_{2}.$
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(d) Show 
\begin_inset Formula $\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v_{3}}$
\end_inset

 is an orthogonal set (hence a basis of 
\begin_inset Formula $\mathbb{R}^{3}$
\end_inset

).
 
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(e) Find the coordinates of 
\begin_inset Formula $\mathbf{v}$
\end_inset

 relative to this basis.
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(10) 
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2.

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 Set up and solve the normal equations for the system 
\begin_inset Formula $A\mathbf{x}=\mathbf{b}$
\end_inset

, where
\begin_inset Formula $A=\left[\mathbf{v}_{1},\mathbf{v}_{2}\right]$
\end_inset

, 
\begin_inset Formula $\mathbf{v}_{1}=\left(1,1,0\right))$
\end_inset

, 
\begin_inset Formula $\mathbf{v}_{2}=\left(-1,1,1\right)$
\end_inset

 and 
\begin_inset Formula $\mathbf{b}=\left(2,1,1\right)$
\end_inset

.
 Is the least squares solution a genuine solution?
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3.

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 (14) Find an eigensystem for the matrix 
\begin_inset Formula $A=\left[\begin{array}{ccc}
1 & 0 & 0\\
-2 & 1 & 0\\
1 & 0 & 1\end{array}\right].$
\end_inset

 Give reasons why the matrix 
\begin_inset Formula $A$
\end_inset

 is or is not diagonalizable.
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(24) 
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4.

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 Matrix 
\begin_inset Formula $A=\left[\begin{array}{cc}
-2 & 2\\
2 & 1\end{array}\right]$
\end_inset

 has eigenvalues 
\begin_inset Formula $-3$
\end_inset

, 
\begin_inset Formula $2$
\end_inset

, and eigenvectors 
\begin_inset Formula $\mathbf{v}_{1}=\left(2,-1\right)$
\end_inset

, 
\begin_inset Formula $\mathbf{v}_{2}=\left(1,2\right)$
\end_inset

.
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\begin_layout Standard
(a) Use this information to find a diagonalizing matrix 
\begin_inset Formula $P$
\end_inset

 for 
\begin_inset Formula $A$
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 and resulting diagonal matrix 
\begin_inset Formula $D$
\end_inset

.
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(b) Use (a) to find a formula for powers of 
\begin_inset Formula $A$
\end_inset

 in terms of powers of eigenvalues of 
\begin_inset Formula $A.$
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(c) Find unit vectors 
\begin_inset Formula $\mathbf{u}_{1}$
\end_inset

, 
\begin_inset Formula $\mathbf{u}_{2}$
\end_inset

 in the directions of 
\begin_inset Formula $\mathbf{v}_{1}$
\end_inset

, 
\begin_inset Formula $\mathbf{v}_{2}$
\end_inset

, respectively, and exhibit an orthogonal matrix 
\begin_inset Formula $U$
\end_inset

 that diagonalizes 
\begin_inset Formula $A$
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.
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 (18) 
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5.

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 Fill in the blanks, or answer T/F:
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(a) If 
\begin_inset Formula $A$
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 is a real matrix, then 
\begin_inset Formula $A^{T}A$
\end_inset

 is symmetric (T/F) 
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.
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(a) Eigenvalues of a matrix cannot be zero (T/F) 
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.
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(b).
 If 
\begin_inset Formula $Q$
\end_inset

 is an 
\begin_inset Formula $n\times n$
\end_inset

 orthogonal matrix and 
\begin_inset Formula $\mathbf{v}\in\mathbb{R}^{n},$
\end_inset

 then 
\begin_inset Formula $\left\Vert Qv\right\Vert =\left\Vert v\right\Vert $
\end_inset

 (T/F)
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.
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(c) If 
\begin_inset Formula $\rho(A)<1$
\end_inset

 and 
\begin_inset Formula $\mathbf{x}^{(k+1)}=A\mathbf{x}^{(k)},$
\end_inset

 then 
\begin_inset Formula $\lim_{k\rightarrow\infty}\mathbf{x}^{(k)}$
\end_inset

 equals 
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.
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(d) The matrix 
\begin_inset Formula $\frac{1}{\sqrt{3}}\left[\begin{array}{cc}
1+i & i\\
i & 1-i\end{array}\right]$
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 is not unitary (T/F) 
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.
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(e) The matrix 
\begin_inset Formula $\left[\begin{array}{ccc}
2 & 1 & 0\\
0 & 1 & 1\\
0 & 0 & 0\end{array}\right]$
\end_inset

 is diagonalizable because 
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.
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(f) The matrix 
\begin_inset Formula $\left[\begin{array}{cc}
1 & i\\
-i & 3\end{array}\right]$
\end_inset

 is unitarily diagonalizable because 
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.
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(g) Every orthonormal set of vectors is linearly independent (T/F) 
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.
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 (h) The component of 
\begin_inset Formula $\mathbf{u}=(1,2,0)$
\end_inset

 along the vector 
\begin_inset Formula $\mathbf{v}=(1,1,1)$
\end_inset

 is 
\begin_inset Formula $\operatorname{comp}_{\mathbf{v}}\left(\mathbf{u}\right)=$
\end_inset


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.
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 (9) 
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6.

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 The vector 
\begin_inset Formula $\mathbf{v}_{1}=\left(1,1\right)$
\end_inset

 is an eigenvector for the symmetric matrix 
\begin_inset Formula $A=\left[\begin{array}{cc}
2 & -1\\
-1 & 2\end{array}\right]$
\end_inset

.
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\begin_layout Standard
(a) Find a vector orthogonal to 
\begin_inset Formula $\mathbf{v}_{1}$
\end_inset

 and show it is an eigenvector of 
\begin_inset Formula $A$
\end_inset

.
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 (b) (
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Honors students only
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) For real eigenvectors 
\begin_inset Formula $\mathbf{v}_{1}$
\end_inset

, 
\begin_inset Formula $\mathbf{v}_{2}$
\end_inset

 of a real symmetric matrix 
\begin_inset Formula $A$
\end_inset

 corresponding to (real) eigenvalues 
\begin_inset Formula $\lambda_{1}\neq\lambda_{2}$
\end_inset

, we have 
\begin_inset Formula $\mathbf{v}_{1}^{T}A\mathbf{v}_{2}=\mathbf{v}_{2}^{T}A\mathbf{v}_{1}$
\end_inset

.
 Use this to deduce that 
\begin_inset Formula $\mathbf{v}_{1}^{T}\mathbf{v}_{2}=0$
\end_inset

.
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