Question 1. Given Ax = b, provide solutions for the following systems of equations

Question 2. Let

a) Draw the set X on a two-dimensional plane.

b) Let for all x ∈ X}

Draw the set Yon a two-dimensional plane. Comment on the mapping.

c) for all x ∈ X}.

Draw the set Z on a two-dimensional plane. Comment on the mapping.

Question 3. For each of the following statements, either show that it is true or give a counterexample.

a) If AB is full rank then A and B are full rank.

b) If A and B are full rank then AB is full rank.

Question 4. About Cauchy - Schwartz Inequality:

a) Suppose a ≥ 0, c ≥ 0 and for all λ ∈ R, a + 2bλ + cλ² ≥ 0 . Show that

                   b ≤ √ac.

b) Given u, w ∈ Rⁿ, explain why (u + λw)^T(U + λW) ≥ 0 for all λ ∈ R.

c) Apply a. to the quadratic resulting from the expression in b. , to get the Cauchy-Schwartz Inequality:

Question 5. Find a basis for the null space and a basis for the range space of