Science:Math Exam Resources/Courses/MATH307/April 2012/Question 04 (b)
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Question 04 (b)
Consider a network arranged in the shape of an octahedron as in this diagram:
The nodes have been labeled with the large numbers and 3 of the edges (resistors) have been given orientations and labeled with small numbers. Assume all resistances are equal to 1. Let D be the incidence matrix (for some choice of labeling and arrows for the remaining edges) and let L be the Laplacian.
(b) Write down a non-zero vector in N(DT).
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Science:Math Exam Resources/Courses/MATH307/April 2012/Question 04 (b)/Hint 1
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From part (a) of the question, we determined that so the dimension of is equal to 1, .
To find , we can use the rank-nullity theorem which states: Where m is the number of columns in the matrix.
Remember that so:
So, if we solve for , we get:
We can now use to solve for using the rank-nullity again. Finding does not directly give a solution to the question, but it does tell us how many vectors there are in . The rank-nullity theorem states that: where n is the number of rows. Plugging in for what we solved above , we get:
This shows that there is only 1 vector in
Now solving for what the question asks, lets look at :
To find the nullspace of , we have to find vectors such that . Looking at , we can easily see that the only solution to this problem is:
This also matches our expectation that there is only 1 vector in N(DT).