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Exam\InsetSpace ~
2 
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Math\InsetSpace ~
314
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Fall 2006
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Name:
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 Show your work in the spaces provided below for full credit.
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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., 
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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.
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(30) 
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1.

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 Let 
\begin_inset Formula $A=[\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3},\mathbf{v}_{4}]=\left[\begin{array}{cccc}
1 & 4 & 1 & -1\\
2 & 4 & 1 & -1\\
4 & 8 & 2 & -2\end{array}\right]$
\end_inset

 with reduced row echelon form 
\begin_inset Formula $R=\left[\begin{array}{cccc}
1 & 0 & 0 & 0\\
0 & 1 & \frac{1}{4} & 0\\
0 & 0 & 0 & 0\end{array}\right]$
\end_inset

.
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\begin_layout Standard
(a) Find a basis for 
\begin_inset Formula $\mathcal{R}(A)$
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, the row space of 
\begin_inset Formula $A$
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.
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 (b) Find a basis for 
\begin_inset Formula $\mathcal{C}(A)$
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, the column space of 
\begin_inset Formula $A$
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.
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 (c) Find a basis for 
\begin_inset Formula $\mathcal{N}(A)$
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, the null space of 
\begin_inset Formula $A$
\end_inset

.
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 (d) Find all possible linear combinations of the vectors 
\begin_inset Formula $\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3},\mathbf{v}_{4}$
\end_inset

 that sum to 
\begin_inset Formula $\mathbf{0}.$
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 (e) Which 
\begin_inset Formula $\mathbf{v}_{j}$
\end_inset

's are redundant in the list of vectors 
\begin_inset Formula $\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3},\mathbf{v}_{4}?$
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 (f) Find a basis of 
\begin_inset Formula $\mathbb{R}^{3}$
\end_inset

 containing a basis of 
\begin_inset Formula $\mathcal{C}(A)$
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.
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(16) 
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2.
 
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Use the Subspace Test to decide if 
\begin_inset Formula $W$
\end_inset

 is a subspace of the vector space 
\begin_inset Formula $V$
\end_inset

, where
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(a) 
\begin_inset Formula $V=\mathbb{R}^{3}$
\end_inset

 and 
\begin_inset Formula $W=\left\{ \left(a,b,a-b+1\right)\,|\, a,b\in\mathbb{R}\right\} $
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 (b) 
\begin_inset Formula $V=C[0,1]$
\end_inset

, the continuous functions on 
\begin_inset Formula $[0,1]$
\end_inset

 and 
\begin_inset Formula $W=\left\{ f\left(x\right)\,|\, f\left(x\right)\in C\left[0,1\right]\mbox{ and }f\left(1\right)=0\right\} $
\end_inset

.
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 (10) 
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3.

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 Assume that 
\begin_inset Formula $1+x$
\end_inset

, 
\begin_inset Formula $x+x^{2}$
\end_inset

, 
\begin_inset Formula $1-x$
\end_inset

 is a basis of 
\begin_inset Formula $\mathcal{P}_{2}$
\end_inset

, the space of polynomials of degree at most two, and find the coordinates
 of 
\begin_inset Formula $2+x^{2}$
\end_inset

 relative to this basis.
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(8) 
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4.
 
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Let 
\begin_inset Formula $A=\left[\begin{array}{ccc}
1 & 0 & 3\\
0 & 1 & 0\\
1 & 0 & -1\end{array}\right]$
\end_inset

.
 Find the adjoint matrix 
\begin_inset Formula $\operatorname{adj}\left(A\right)$
\end_inset

 of 
\begin_inset Formula $A$
\end_inset

.
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(12) 
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5.
 
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You are given that 
\begin_inset Formula $\mathbf{w}_{1}=(0,1,0)$
\end_inset

, 
\begin_inset Formula $\mathbf{w}_{2}=(1,1,1)$
\end_inset

 is a linearly independent set in 
\begin_inset Formula $V=\mathbb{R}^{3}$
\end_inset

 and 
\begin_inset Formula $\mathbf{v}_{1}=(1,3,1)$
\end_inset

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

, 
\begin_inset Formula $\mathbf{v}_{3}=(1,0,1)$
\end_inset

 is a basis of 
\begin_inset Formula $V$
\end_inset

.
 Steinitz substitution says that 
\begin_inset Formula $\mathbf{w}_{1}$
\end_inset

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

 can be substituted into the basis in place of certain 
\begin_inset Formula $\mathbf{v}_{i}$
\end_inset

's.
 Which substitutions work?
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(16) 
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6.

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 Fill in the blanks or answer True/False (T/F).
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\begin_layout Standard
(a) Every vector space is finite dimensional (T/F) 
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.
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 (b) Elementary row operations on a matrix do not change the column space
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.
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 (c) If 
\begin_inset Formula $\mathbf{x}=\mathbf{x}_{0}$
\end_inset

 and 
\begin_inset Formula $\mathbf{x}=\mathbf{x}_{1}$
\end_inset

 are both vector solutions to the linear system 
\begin_inset Formula $A\mathbf{x}=\mathbf{b},$
\end_inset

 then 
\begin_inset Formula $\mathbf{x}_{1}-\mathbf{x}_{0}$
\end_inset

 is in the null space of 
\begin_inset Formula $A.$
\end_inset

 (T/F) 
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.
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 (d) The function 
\begin_inset Formula $T:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$
\end_inset

 given by 
\begin_inset Formula $T((x,y))=(x+y,x-2y)$
\end_inset

 is linear (T/F) 
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 and one-to-one (T/F) 
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.
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 (e) The Basis Theorem asserts that every finite dimensional vector space
 
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.
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 (f) The Dimension Theorem asserts that 
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.
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 (g) A basis of a vector space is by definition
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.
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 (10) 
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7.

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 (a) Show that the columns of the matrix 
\begin_inset Formula $\left[\begin{array}{ccc}
1 & 0 & 1\\
1 & 0 & 1\\
3 & 0 & 1\end{array}\right]$
\end_inset

 form a linearly dependent set.
 
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(b) (
\emph on
Honors students only
\emph default
) Prove that any set of vectors 
\begin_inset Formula $\mathbf{v}_{1},\mathbf{v}_{2},\ldots,\mathbf{v}_{n}$
\end_inset

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

 that contains the zero vector is a linearly dependent set.
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