Worksheet 5/19
1. Determine whether or not the set, S = {(x1 , x2 , x3 ) ∈ R3 : x21 + x22 + x23 ≤ 4} is a vector
space over R.
2. Describe the vector space Zn2 = {(x1 , · · · , xn ) : xi ∈ Z2 } geometrically and algebraically
for n = 2, 3.
3. Use the inequality, a2 + b2 ≥ 2ab, to prove Cauchy-Schwarz for R2 . Hint: squaring both
sides it suffices to show that (⟨x, y⟩)2 ≤ ||x||2 ||y||2
4. Without using the Law of cosines, show that if ||x|| = ||y|| = 1 and ⟨x, y⟩ = 1, then
x = y. Hint: Show that x − y = 0 by showing that ||x − y||2 = 0
5. Let c ∈ R3 and b ∈ R be fixed. Describe the following regions in R3 and determine
whether or not they are a vector space over R.
a. cT x = b
b. xT x = b2