Define two planes in r3 give a geometric explanation for

Question

1. Let V = M^2x2 (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.