Q1: write down the answer and prove it.

Q2: Prove Cauchy-Schwartz inequality in two ways.

Q3: Let be a matrix of size . Denote the range space of as and the null space of , repectively. Prove .

Q4: For any two matrices, prove .

Q5: Prove a useful inequality: .

Q6: Define .Compute and .

Q7: Define . Compute and .

Q8: For the logistic regression example in lecture notes, compute .

Q9: Define . Prove.

Q10: Find at least one example in either of the following two fields that can be formulated as an optimization problem and show how to formulate it