Problem solving seminars IX and X
31. Let f, g : [0, 1] → [0, ∞) be continuous functions safisfying
sup{f(x); x ∈ [0, 1]} = sup{g(x); x ∈ [0, 1]}.
Prove that there exists t ∈ [0, 1] with
f2
(t) + 3f(t) = g2
(t) + 3g(t).
32. Let R be the ring of matrices of the form
a b
0 c
.
What are all 2-sided ideals in R?
33. Let A be the 3 × 3 matrix


1 −1 0
−1 2 −1
0 −1 1

 .
Determine all real numbers a for which
lim
n→∞
an
An
exists and is nonzero as a matrix.
34. Let E be the set of all continuous functions u : [0, 1] → R satisfying
|u(x) − u(y)| ≤ |x − y| for x, y ∈ [0, 1], u(0) = 0.
Let ϕ : E → R be defined by
ϕ(u) =
1
0
(u(x)2
− u(x))dx.
Show that ϕ achieves its maximum value on some element of E.
35. Let Mn×n(F) be the ring of n × n matrices over a field F. Prove that it has no
2-sided ideals except Mn×n(F) and {0}.
36. Find all left ideals of the ring Mn×n(F).
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