Homework # 1 - Math 241
The problems on this homework must be done alone. The honor code is in effect.
First name: Last name:
1. In the x-y plane, describe the following: \(2x + y = 1\) and \(2x + 2y = 0\).
a. They are parallel lines.
b. They represent two planes intersecting in a line.
c. They are the same line.
d. They are intersecting lines.
e. They are parallel planes.
2. Given three planes, which of the following can't be true of the mutual intersection of all three?
a. The intersection is empty.
b. The intersection consists of one line.
c. The intersection consists of two points.
d. The intersection consists of a plane.
e. The intersection consists of one point.
In problems 3-5, consider the matrix $$\left[ \begin{array}{ccc} 1 & 2 & 3 \\ 3 & 6 & 12 \\ 3 & 3 & 9 \end{array} \right]$$.
3. What elementary row operation transforms the given matrix to the matrix $$\left[ \begin{array}{ccc} 1 & 2 & 3 \\ 1 & 2 & 4 \\ 3 & 3 & 9\end{array} \right]$$
a. Replace row 2 by row 3 minus row 2.
b. Replace row 2 by row 2 minus 2 times row 1.
c. Multiply row 2 by \(\frac{1}{3}\).
d. Replace row 2 by 2 times row 1.
e. Multiply row 3 by 1.
4. What elementary row operation transforms the given matrix to the matrix $$\left[ \begin{array}{ccc} 1 & 2 & 3 \\ 1 & 2 & 6 \\ 3 & 3 & 9\end{array} \right]$$
5. What is the reduced echelon form of the given matrix?
a. $$\left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right]$$
b. $$\left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 2\\ 0 & 0 & 0 \end{array} \right]$$
c. $$\left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 3 \\ 0 & 0 & 0 \end{array} \right]$$
d. $$\left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right]$$
e. $$\left[\begin{array}{ccc} 1 & 2 & 3 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right]$$