Lab Project 1 (Due 2/5/03)

Date: MATH 2270-1 - Spring 2003

1. Each linear equation in $3$ unknowns represents a plane in $R^3$. A system of linear equations in $R^3$ is, then, represented by intersecting planes in $R^3$. Illustrate this process for the following systems of equations. In each case, additionally, find the solution.

   1. $$\begin{cases}x+2y+z =0 \\ 2x-2y+z=3\\ x+2z=1 \end{cases}.$$

      
      

   2. $$\begin{cases}x+y+2z =1 \\ x+z=0\\ y+z=0 \end{cases}.$$

      
      

2. Figure 1 shows an arrangement of wires and nodes. It is known that the temperature at each node is the average of the temperature at the nodes directly connected. For example, if $P_1$ denotes the temperature at the point $P_1$, we have

   $$P_1 = \frac{T_1 + P_2 + P_6 + P_5}{4}.$$

   
   
   1. Write down the equations that express all the temperature relations corresponding to the scheme shown in Figure 1.

      Figure 1: Temperature grid

      
   2. Suppose that $T_1$, $T_2$, $T_3$ and $T_4$ are given. Find an expression for each of the temperatures $P_1,\ldots, P_6$.
   3. What are the temperatures at the $P$-nodes if $T_1=T_3=0$, $T_2=100$ and $T_4 = 200$?
   4. Write down formulas for the temperatures at the $P$-nodes when $T_1=T_3=0$ and $T_2=100$. According to your formulas, which node is the most sensitive to the temperature $T_4$? Does this agree with your intuition?
   5. Write down the equations that express all the temperature relations corresponding to the scheme shown in Figure 2.

      Figure 2: Temperature grid

      
   6. Solve the system of equations of the previous item. Is there a unique solution?

3. Analysis of a linear transformation on $R^2$.
   1. Design a ``test graph'' according to Figure 3

      Figure 3: test graph

      
   2. Let $T:R^2\rightarrow R^2$ be the rotation by $30$ degrees in the clockwise direction. Write the coefficient matrix of $T$.
   3. Apply $T$ to the test graph to verify the effect graphically.
   4. Describe the effect on the ``test graph'' of the linear transformation whose coefficient matrix is

      $$A =\left( \begin{array}{cc} 0 & 1\\ 1 & 0 \end{array} \right).$$

      
      

4. Given

   $$A = \left( \begin{array}{ccccc} 2 & 3 & 0 & -1 & k\\ 0 & 1 & 7 & ... ... & 2 & 1 & 0 & 1\\ 7 & 2 & 3 & 1 & 1\\ 0 & 0 & 1 & -1 & -1 \end{array} \right),$$

   
   
   find all the values of $k$ such that $A$ is invertible.

5. Given the vectors

   $$v = \left( \begin{array}{c} 12\\ -3\\ -8\\ 8\\ 40 \end{array} \ri... ...) ,\quad w_3 = \left( \begin{array}{c} 1\\ -1\\ 3\\ 0\\ 4 \end{array} \right) ,$$

   
   
   is it possible to write $v$ as a linear combination of $w_1$, $w_2$ and $w_3$? If the answer is positive, find the coefficients of the linear combination.