Homework #3

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Q1 (6 points)

Prove that is a basis for by finding a basis of for which it is the dual basis.
Q2 (6 points)

For a positive integer , determine the set of ordered pairs

(Hint: You may find it helpful to research `Sylvester's rank inequality'; if you use it, sketch a proof in your
solution.)
Q3 (6 points)
Let , and suppose . If the solution sets and are nonempty and equal, does it follow that ? Explain.

Q4 (6 points)

Given a matrix , let denote the matrix obtained from by `rotating' it clockwise. For
example,
Find (with proof) a formula for in terms of (for arbitrary size ).
Q5 (6 points)

(a) Compute the block matrix product

(b) Find (with proof) a formula for .