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.
![732_matrix.jpg](https://secure.tutorsglobe.com/CMSImages/732_matrix.jpg)
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.