Homework # 3 - Math 241
The problems on this homework must be done alone. The honor code is in effect.
First name: Last name:
1. If the vectors $$ \vec{u}= \left[ \begin{array}{c} 1\\-2\\3 \end{array}\right], \hspace{.2 in} \vec{v}= \left[ \begin{array}{c} -1\\5\\1 \end{array}\right], \hspace{.2 in} \vec{w}= \left[ \begin{array}{c} 1\\1\\h \end{array}\right]$$ are linearly dependent, find the value of \(h\).
a. 0
b. 2
c. 5
d. 7
e. 11
For problems 2 and 3, say \(\vec{u}, \vec{v}\), and \(\vec{w}\) are linearly dependent vectors in \(R^3\).
2. The statement: the vectors \(\vec{u}+\vec{v}, \vec{v}\), and \(2\vec{w}\) form a linearly independent set is
a. Definitely false
b. Definitely true
c. There is not enough information to say.
3. The statement: the vectors \(\vec{u}, \vec{v}\), and \(\vec{w}\) span \(R^3\) is
4. If \(A\) is a 3 by 5 matrix, and the equation \(A{\bf x}= {\bf b}\) where \({\bf b}\) is a vector in \({\bf R}^3\) has no solutions, what can you say about the number of pivot positions of \(A\)?
a. There are none.
b. There are at most 2.
c. There are at least 2.
d. There are exactly 3.
e. There are 5.
5. If \(A\) is a 3 by 5 matrix, and the equation \(A{\bf x}= {\bf b}\) has a solution for all vectors \({\bf b}\) in \({\bf R}^3\), what can you say about the number of pivot positions of \(A\)?
6. If \(A\) is an m by n matrix (m rows and n columns), and the equation \(A{\bf x}= {\bf b}\) where \({\bf b}\) is a vector in \({\bf R}^n\) has exactly one solution, what can you say about m and n?
a. \(m \leq n\).
b. \(m \geq n\).
c. \(m < n\).
d. \(m > n\).
e. You cannot draw any conclusions about m and n.
7. Say \({\bf a_1}, {\bf a_2}, \ldots, {\bf a_5}\) are vectors in \({\bf R}^7\), then
a. the vectors definitely don't span \({\bf R}^7\).
b. the vectors definitely do span \({\bf R}^7\).
c. there is not enough information to determine whether or not the vectors span \({\bf R}^7\).
8. Say \({\bf a_1}, {\bf a_2}, \ldots, {\bf a_7}\) are vectors in \({\bf R}^5\), then
a. the vectors definitely don't span \({\bf R}^5\).
b. the vectors definitely do span \({\bf R}^5\).
c. there is not enough information to determine whether or not the vectors span \({\bf R}^5\).
9. Do the vectors $$ {\bf a_1} = \left[\begin{array}{c} 1 \\ 1 \\ 1 \end{array}\right], {\bf a_2} = \left[\begin{array}{c} 1 \\ 2 \\ 3 \end{array} \right], {\bf a_3} = \left[\begin{array}{c} 3 \\ 2 \\ 1 \end{array} \right], {\bf a_3} = \left[\begin{array}{c} -1 \\ 1 \\ 3 \end{array} \right]$$ span \({\bf R}^3\)?
a. No.
b. Yes.
c. there is not enough information.