Science:Math Exam Resources/Courses/MATH307/December 2006/Question 02 (a)
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Question 02 (a)
Consider the matrix
(a) Find a basis for the nullspace of A.
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
To find the basis of null space of A, we solve
This can be done by finding the reduced row echelon form of A as follows:
First, subtract the first row twice from the second row and once from the first row.
Then, divide the second row by 3 and add it to the third row
From this we can see that column 1 and column 3 are the only pivot columns and so the free variables are x2, x4 and x5, while the basic variables are x1 and x3
Solving this r.r.e.f of A for homogeneous solution gives us
Note: we can use matlab command null (rref(A), ’r’) to directly calculate the basis of null space for this matrix A