1. Let V = M2x2 (C) be the vector space of all complex (2 x 2)-matrices. Are the following three matrices linearly independent?
|
1 |
0 |
|
|
0 |
i |
|
|
0 |
0 |
| A1 = |
0 |
0 |
|
A2 = |
0 |
0 |
|
A3 |
1+i |
0 |
2. Suppose we have a thin 3 x 3 square metal plate whose edges are kept at fixed temperatures, as shown in the figure below.
We are interested to find the temperatures, x1, x2, x3, and x4, at the four interior points, using the Mean Value
Property for Heat Conduction: The temperature at any interior point is the average of the temperature of its neighboring points.
a) The Mean Value Property says that
x1 = 1/4 (x2 + x3 + 1)
Similarly, write the other 3 equations, for x2, x3, and x4.
b) Solve the system of 4 equations in part (a). Hint: first multiply every equation by 4.
3. The equations
a1x + b1y+ c1z = 0
a2x + b2y + c2z = 0
define two planes in R3. Give a geometric explanation for why there must be an infinite number of solutions.