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{\bf Exam~2 \hfill Math~314 \hfill Sample~Exam}
{\bf Name:}\rule{2in}{.01in} \hfill \hfill {\bf Score:}\rule{1in}{.01in}
{\hspace{.25in} \it Instructions:\/} Show your work in the spaces
provided below for full credit. Use the reverse side for additional
space, {\it but clearly so indicate.\/} Identify answers by underlining
or circling, where appropriate. Point values of problems are in
parentheses. Notes, calculators or text not allowed. \newline
\rule[.2in]{7in}{.01in}
(26) 1. In the following you are given that
$$ A = [v_1,v_2,v_3,v_4,v_5] = \left[ \begin{array}{ccccc}
1 & -1 & 5 & 1 & 0 \\
2 & 1 & 4 & 2 & 1 \\
3 & 0 & 9 & 2 & 5 \\
-1 & -1 & -1 & -1 & 2
\end{array} \right]
\mbox{ has Row Echelon Form } E = \left[ \begin{array}{ccccc}
1 & 0 & 3 & 0 & -1 \\
0 & 1 & -2 & 0 & -5 \\
0 & 0 & 0 & 1 & 4 \\
0 & 0 & 0 & 0 & 0
\end{array} \right].
$$
(a) What is the rank of $A$ and dimensions of ${\cal R}(A)$, ${\cal C}(A)$ and ${\cal N}(A)$?
\vspace{0.1in}
(b) Find a basis for the row space of $A$.
\vspace{0.1in}
(c) Find a basis for the column space of $A$.
\vspace{0.1in}
(d) Find a basis for the null space of $A$.
\vspace{0.1in}
(e) Express $v_5$ as a linear combination of $v_1, v_2, v_3, v_4$.
(12) 2. Find the inverse of the following matrix by using its adjoint
matrix: $\left[ \begin{array}{ccc} 2 & 2 & 1 \\ 1 & 2 & 0 \\ 1 & 0 & 0
\end{array} \right]$
\vspace{0.1in}
(12) 3. In the following, $u_1 = [1,0,1]^T$ and $u_2 = [1,-1,1]^T$.
(a) Show that $u_1, u_2$ form a linearly independent set.
\vspace{0.1in}
(b) Does $v = [2,1,2]$ belong to the space $\mbox{lin}\{u_1,u_2\}$? Give reasons for your answer.
\vspace{0.1in}
(c) Find a basis of $R^3$ which contains $u_1$ and $u_2$. Give
reasons for your answer.
\vspace{0.2in}
(14) 4. Use the Subspace Test to decide if $W$ is a subspace of the
given vector space $V$.
(a) $V$ is the space of all 2x2 matrices over the reals with the usual
matrix addition and scalar multiplication and $W$ is the set of 2x2
matrices of the form $\{ \left[ \begin{array}{cc} a & 0 \\ 0 & b
\end{array} \right] | a, b$ are reals\}.
\vspace{0.1in}
(b) $V$ is the space $C[0,1]$ of continuuous functions on [0,1] and W is
set of $f$ in $C[0,1]$ such that $f(0) = 2$.
\vspace{0.2in}
(12) 5. Suppose that the linear system $Ax = b$ is a consistent system
of equations, where $A$ is an mxn matrix and $x = [x_1, ..., x_n]^T$.
Prove the following:
(a) $b \in {\cal C}(A)$.
\vspace{0.1in}
(b) If $x_0$ is any particular solution to the system, then every vector
of the form $x_0 + z$, where $z \in {\cal N}(A)$, is a solution to the
system.
\vspace{0.1in}
(c) If the set of columns of $A$ has redundant vectors in it, then the
system has more than one solution.
\vspace{0.2in}
(24) 6. Fill in the blanks or answer T/F:
(i) Every spanning set of a vector space contains a basis of the space
(T/F) \rule{1in}{.01in}.
\vspace{.1in}
(ii) Every vector space is finite dimensional (T/F) \rule{1in}{.01in}.
\vspace{.1in}
(iii) The dimension of $C^3$ as a vector space over $R$ is 6 (T/F)
\rule{1in}{.01in}.
\vspace{.1in}
(iv) $\det AB = \det A \det B$ (T/F) \rule{1in}{.01in}.
\vspace{.1in}
(v) $\det (A+B) = \det A + \det B$ (T/F) \rule{1in}{.01in}.
\vspace{.1in}
(vi) The set consisting of the 0 vector is a linearly independent set
(T/F) \rule{1.3in}{.01in}.
\vspace{.1in}
(vi) The vectors $v_1, v_2, ... ,v_n$ are linearly dependent means that
\rule{5in}{.01in}
\vspace{.1in}
(iv) If $u = [1,2,-1,1]$ and $v = [-2,1,0,0]^T$, then $||u|| = $
\rule{1in}{.01in} and $u\cdot v = $ \rule{1in}{.01in}.
\vspace{.1in}
(v) In terms of $A$, the determinant of $-2A$, where $A$ is $n \times
n$, is \rule{1in}{.01in}.
\vspace{.1in}
(vi) The vectors $[1,0]^T$, $[0,1]^T$ and $[1,1]^T$ are linearly
\rule{1.6in}{.01in}.
\vspace{.1in}
(vii) The dimension of the real vector space $R^n$ is \rule{1in}{.01in}.
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