Science:Math Exam Resources/Courses/MATH307/December 2012/Question 06 (f)
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Question 06 (f) 

Suppose that A is a matrix with singular value decomposition where (f) Let where U and V are the matrices given above. Then is a matrix with a nontrivial null space. What is a basis for ? What is the norm ? 
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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

To find the nullspace , look for vectors x such that 
Hint 2 

Since U is unitary, the only way that is that . Find the nullspace of and relate the result to . Solve for x. 
Hint 3 

To calculate use that . 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Let's start by calculating the nullspace of , that is, we are looking for vectors x such that Since U is unitary, Uy = 0 implies y = 0. Hence, for is only possible if and only if Let's abbreviate . Then, in order to find the nullspace of we are looking for vectors z such that This equation can quickly be solved: , for any real value of t. Since we solve for x and find Therefore, Next, observe that Again following the logic of part (c) it holds that 