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\language english
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Final\InsetSpace ~
Exam 
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314
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Sample
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Name:
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Score:
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Instructions:
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\shape default
 Show your work in the spaces provided below for full credit.
 Use the reverse side for additional space, 
\shape italic
but clearly so indicate.
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\shape default
 You must clearly identify answers and show supporting work to receive any
 credit.
 Exact answers (e.g., 
\begin_inset Formula $\pi$
\end_inset

) are preferred to inexact (e.g., 
\begin_inset Formula $3.14$
\end_inset

).
 Point values of problems are given in parentheses.
 Notes or text in 
\emph on
any
\emph default
 form not allowed.
 Calculator is allowed.
 
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(20) 
\series bold
1.

\series default
 The system of equations 
\begin_inset Formula \begin{eqnarray*}
x_{1}+x_{2} & = & 2\\
x_{3}+x_{4} & = & 1\\
2x_{1}+4x_{2}+2x_{4} & = & 6\end{eqnarray*}

\end_inset

 has coefficient matrix 
\begin_inset Formula $A$
\end_inset

 and right hand side 
\begin_inset Formula $\mathbf{b}$
\end_inset

 such that the row-reduced echelon form of 
\begin_inset Formula $[A|\mathbf{b}]$
\end_inset

 is 
\begin_inset Formula $\left[\begin{array}{cccccc}
1 & 2 & 0 & 0 & | & 2\\
0 & 0 & 1 & 0 & | & 0\\
0 & 0 & 0 & 1 & | & 1\end{array}\right]$
\end_inset

.
 Use this information to answer the following:
\end_layout

\begin_layout Standard
(a) Find a basis for the null space of 
\begin_inset Formula $A$
\end_inset

.
\end_layout

\begin_layout Standard
(b) Find the form of a general solution of the system 
\begin_inset Formula $A\mathbf{x}=\mathbf{b}$
\end_inset

.
\end_layout

\begin_layout Standard
(c) Find a basis for the row space of 
\begin_inset Formula $A$
\end_inset

.
\end_layout

\begin_layout Standard
(d) No matter what the right hand side of 
\begin_inset Formula $\mathbf{b}$
\end_inset

 is, this system has solutions.
 In terms of rank, why do we know this?
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 (12) 
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2.

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 Let 
\begin_inset Formula $A=\left[\begin{array}{ccc}
0 & 1 & 0\\
2 & 0 & 1\\
2 & 0 & 2\end{array}\right]$
\end_inset

.
\end_layout

\begin_layout Standard
(a) Find 
\begin_inset Formula $A^{-1}$
\end_inset

.
\end_layout

\begin_layout Standard
(b) Use (a) to solve the equation 
\begin_inset Formula $A\mathbf{x}=\left[\begin{array}{c}
2\\
1\\
1\end{array}\right]$
\end_inset

 for 
\begin_inset Formula $\mathbf{x}$
\end_inset


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 (22) 
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3.

\series default
 Let 
\begin_inset Formula $A=\left[\begin{array}{ccc}
1 & 0 & 1\\
-1 & 1 & 1\\
2 & 1 & 4\end{array}\right]$
\end_inset

.
\end_layout

\begin_layout Standard
(a) Find the reduced row-echelon form and rank of 
\begin_inset Formula $A$
\end_inset

.
\end_layout

\begin_layout Standard
(b) Find a basis for the column space of 
\begin_inset Formula $A$
\end_inset

.
\end_layout

\begin_layout Standard
(c) Determine which of the following vectors is in the column space of 
\begin_inset Formula $A$
\end_inset

 and, if so, express the vector as a linear combination of the columns of
 
\begin_inset Formula $A$
\end_inset

:
\end_layout

\begin_layout Standard
\begin_inset Formula $b_{1}=[2,1,0]^{T}$
\end_inset

, hspace.3in 
\begin_inset Formula $b_{2}=[2,-3,3]^{T}$
\end_inset

.
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 (14) 
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4.

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 Use the Gram-Schmidt process on the basis
\end_layout

\begin_layout Standard
\begin_inset Formula $u_{1}=\left[\begin{array}{c}
1\\
0\\
1\\
0\end{array}\right]$
\end_inset

, 
\begin_inset Formula $u_{2}=\left[\begin{array}{c}
1\\
1\\
1\\
0\end{array}\right]$
\end_inset

, and 
\begin_inset Formula $u_{3}=\left[\begin{array}{c}
0\\
0\\
1\\
1\end{array}\right]$
\end_inset

 of the subspace 
\begin_inset Formula $W$
\end_inset

 of 
\begin_inset Formula $\mathbb{R}^{4}$
\end_inset

 to produce an orthonormal basis of 
\begin_inset Formula $W$
\end_inset

.
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 (12) 
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5.

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 Calculate the following determinant:
\end_layout

\begin_layout Standard
\begin_inset Formula $\left|\begin{array}{cccc}
1 & 1 & 1 & 1\\
1 & 2 & 3 & 4\\
1 & 3 & 5 & 8\\
1 & 4 & 9 & 9\end{array}\right|$
\end_inset


\end_layout

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 (12) 
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6.

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 The set 
\begin_inset Formula $\left\{ 1,x-\frac{1}{2},x^{2}-x+\frac{1}{6}\right\} $
\end_inset

 is an orthogonal basis of the inner product space 
\begin_inset Formula $\mathcal{P}_{2}$
\end_inset

 of polynomials of degree at most 
\begin_inset Formula $2$
\end_inset

 with the inner product 
\begin_inset Formula $\left\langle p,q\right\rangle =\int_{0}^{1}p\left(x\right)q\left(x\right)dx$
\end_inset

.
 Assume this and find the coordinates of 
\begin_inset Formula $p\left(x\right)=x^{2}$
\end_inset

 with respect to this basis.
 What is the angle between 
\begin_inset Formula $p\left(x\right)$
\end_inset

 and 
\begin_inset Formula $q\left(x\right)=1$
\end_inset

 in this inner product space?
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\end_inset

 (12) 
\series bold
7.

\series default
 Find a least squares solution to the inconsistent system 
\begin_inset Formula \[
\left[\begin{array}{cc}
1 & -1\\
1 & 1\\
1 & 2\end{array}\right]\left[\begin{array}{c}
x_{1}\\
x_{2}\end{array}\right]=\left[\begin{array}{c}
1\\
1\\
3\end{array}\right]\]

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 (16) 
\series bold
8.

\series default
 Let 
\begin_inset Formula $A=\left[\begin{array}{ccc}
2 & 1 & 1\\
0 & 2 & 1\\
0 & 1 & 2\end{array}\right]$
\end_inset

.
\end_layout

\begin_layout Standard
(a) Find all eigenvalues of 
\begin_inset Formula $A$
\end_inset

.
\end_layout

\begin_layout Standard
(b) Find a basis for the eigenspace corresponding to each eigenvalue of
 
\begin_inset Formula $A$
\end_inset

.
\end_layout

\begin_layout Standard
(c) Produce an invertible matrix 
\begin_inset Formula $P$
\end_inset

 and diagonal 
\begin_inset Formula $D$
\end_inset

 such that 
\begin_inset Formula $P^{-1}AP=D$
\end_inset

.
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 (16) 
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9.

\series default
 Let 
\begin_inset Formula $A=\left[\begin{array}{cc}
1 & 1+i\\
1-i & 2\end{array}\right]$
\end_inset

.
 One of the eigenvalues of 
\begin_inset Formula $A$
\end_inset

 is 0.
\end_layout

\begin_layout Standard
(a) Find the eigenvalues of 
\begin_inset Formula $A$
\end_inset

.
\end_layout

\begin_layout Standard
(b) Find a basis for the eigenspace corresponding to each eigenvalue of
 
\begin_inset Formula $A$
\end_inset

.
\end_layout

\begin_layout Standard
(c) Produce a unitary matrix 
\begin_inset Formula $U$
\end_inset

 and diagonal matrix 
\begin_inset Formula $D$
\end_inset

 such that 
\begin_inset Formula $U^{*}AU=D.$
\end_inset


\end_layout

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 (15) 
\series bold
10.

\series default
 Let 
\begin_inset Formula $v$
\end_inset

 be a unit column vector in 
\begin_inset Formula $\mathbb{R}^{n}$
\end_inset

 and 
\begin_inset Formula $H=I_{n}-2vv^{T}$
\end_inset

.
\end_layout

\begin_layout Standard
(a) What is the size of the matrix 
\begin_inset Formula $H$
\end_inset

?
\end_layout

\begin_layout Standard
(b) Give the definition of symmetric matrix and prove 
\begin_inset Formula $H$
\end_inset

 is symmetric.
\end_layout

\begin_layout Standard
(c) Give the definition of orthogonal and prove 
\begin_inset Formula $H$
\end_inset

 is orthogonal.
 
\end_layout

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 (21) 
\series bold
11.

\series default
 Circle T for true, F for false or do not answer.
 Each correct answer is worth 3 points, incorrect answer worth -1 points
 and no answer worth 0, for a minimum of 0 and maximum of 21 points.
\end_layout

\begin_layout Standard
T F (a) If 
\begin_inset Formula $\mathbf{u}$
\end_inset

 and 
\begin_inset Formula $\mathbf{v}$
\end_inset

 are elements of the real inner product space 
\begin_inset Formula $V$
\end_inset

, then 
\begin_inset Formula $\left\langle \mathbf{u},\mathbf{v}\right\rangle <\left\langle \mathbf{v},\mathbf{v}\right\rangle \geq\left\langle \mathbf{u},\mathbf{v}\right\rangle ^{2}$
\end_inset

.
\end_layout

\begin_layout Standard
T F (b) Every real matrix is similar to a diagonal matrix.
\end_layout

\begin_layout Standard
T F (c) Every orthogonal set of vectors is linearly independent
\end_layout

\begin_layout Standard
T F (e) For 
\begin_inset Formula $n\times n$
\end_inset

 matrices 
\begin_inset Formula $A$
\end_inset

 and 
\begin_inset Formula $B$
\end_inset

, 
\begin_inset Formula $(AB)^{*}=A^{*}B^{*}$
\end_inset

.
 
\end_layout

\begin_layout Standard
T F (f) If the linear system 
\begin_inset Formula $Ax=B$
\end_inset

 has a unique solution and 
\begin_inset Formula $A$
\end_inset

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

 matrix, then 
\begin_inset Formula $n\leq m$
\end_inset

.
\end_layout

\begin_layout Standard
T F (g) If 
\begin_inset Formula $V=\operatorname{span}\left\{ \mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3}\right\} $
\end_inset

 and 
\begin_inset Formula $\operatorname{dim}(V)=2$
\end_inset

, then 
\begin_inset Formula $\left\{ \mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3}\right\} $
\end_inset

 is a linearly dependent set.
\end_layout

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 (28) 
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12.

\series default
 Fill in the blank or give a short answer in the following:
\end_layout

\begin_layout Standard
(a) 
\begin_inset Formula $\left[\begin{array}{cc}
1 & 2\\
0 & 1+i\\
0 & 0\end{array}\right]$
\end_inset

 
\begin_inset Formula $\left[\begin{array}{ccc}
1 & 2 & 0\\
0 & i & 1\end{array}\right]=$
\end_inset


\end_layout

\begin_layout Standard
(b) A set of vectors 
\begin_inset Formula $\mathbf{v}_{1},\mathbf{v}_{2},...,\mathbf{v}_{n}$
\end_inset

 in the vector space 
\begin_inset Formula $V$
\end_inset

 is defined to be linearly independent if:
\end_layout

\begin_layout Standard
(c) If 
\begin_inset Formula $A$
\end_inset

 is symmetric matrix, what can you say about the eigenvalues of 
\begin_inset Formula $A$
\end_inset

?
\end_layout

\begin_layout Standard
(d) Find two 
\begin_inset Formula $2\times2$
\end_inset

 matrices 
\begin_inset Formula $A$
\end_inset

 and 
\begin_inset Formula $B$
\end_inset

 such that 
\begin_inset Formula $AB=0$
\end_inset

.
\end_layout

\begin_layout Standard
(f) Let 
\begin_inset Formula $\mathbf{u}=[1,2,1]^{T}$
\end_inset

 and 
\begin_inset Formula $\mathbf{v}=[1,-1,0]$
\end_inset

.
 Then the projection of 
\begin_inset Formula $\mathbf{u}$
\end_inset

 along 
\begin_inset Formula $\mathbf{v}$
\end_inset

 is 
\begin_inset Formula $\mathbf{p}$
\end_inset

 and 
\begin_inset Formula $\mathbf{u}$
\end_inset

 can be written as 
\begin_inset Formula $\mathbf{p}+\mathbf{x}$
\end_inset

 where 
\begin_inset Formula $\mathbf{x}$
\end_inset

 is orthogonal to 
\begin_inset Formula $\mathbf{v}$
\end_inset

.
 Find 
\begin_inset Formula $\mathbf{p},\mathbf{x}$
\end_inset

.
\end_layout

\begin_layout Standard
(g) If 
\begin_inset Formula $W$
\end_inset

 is the row space of the matrix 
\begin_inset Formula $\left[\begin{array}{ccc}
1 & 2 & 0\\
0 & 1 & 1\end{array}\right]$
\end_inset

 then 
\begin_inset Formula $W^{\perp}=$
\end_inset

.
\end_layout

\begin_layout Standard
(h) If 
\begin_inset Formula $\mathbf{u}=\left(-2,1+i,3,1\right)\in\mathbb{C}^{4}$
\end_inset

, then 
\begin_inset Formula $\left\Vert \mathbf{u}\right\Vert _{1}$
\end_inset

 and 
\begin_inset Formula $\left\Vert \mathbf{u}\right\Vert _{\infty}$
\end_inset

 are equal to: 
\end_layout

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