91 academic year Part I.
1.
(i) Let a = 0.9999 . . . = 0.¯9 and b = 1. Is a < b? Or is a = b? Explain your answer.
(ii) Let f : R2 → R be defined by f (x, y) = xy. Show that f is a continuous
function by using the ²-δ language. (10%)
2.
(i) Let f : [0, ∞) → R be a bounded continuous function that is improper Riemann integrable on the interval [0, ∞). Is limx→∞f (x) = 0? Why or why not?
(ii) Let f : R → R be a continuous function. Is f (K) closed when K is a closed subset of R? Is f (M ) bounded and closed when M is a bounded and closed
subset of R? Why or why not? (10%)
3.
f (x) =
½x + 2x2sin(1/x), if x 6= 0;
0, if x = 0.
Show that f0(0) 6= 0 but that f is not locally invertible near 0. Why does this not
contradict the inverse function theorem? (10%)
4. Let ak be a sequence of real numbers. Suppose that the series P∞
k=0ak converges.
(i) Does the power series P∞
k=0akxk converge uniformly on the interval [0, 1]?
Why or why not?
(ii) Is limx→1−P∞
k=0akxk=P∞
k=0ak? Why or why not? (10%)
5. Define ρ: R2× R2 → R by ρ((x1, y1), (x2, y2)) = max{|x1− x2|, |y1− y2|}.
(i) Check that (R2, ρ) is a metric space.
(ii) Let d be the usual metric of R2 i.e., d((x1, y1), (x2, y2)) = ((x1− x2)2+ (y1− y2)2)1/2. Is an open set in (R2, d) also an open set in (R2, ρ)? Why or why
not? (10%)
Part II.
6. Does there exist 3 × 3 matrices A and B satisfying AB − BA = I (I is the
identity matrix)? Why? (8%)
7. Give an example of two 3 × 3 matrices which are similar, but not
unitarily equivalent, and explain your answer. (8%)
8. Let V be a vector space and T : V → V be linear. Show that if T2 = T ,
then V = ker(T ) ⊕ ran(T ), the direct sum of kernel and range of T . (8%) 9. Suppose A, B and C are 3 × 3 matrices. Prove that
det
µ·A B
0 C
¸¶
= det(A) det(C).
(8%) 10. Let (V, <, >) be a finite-dimensional inner product space over C and
F : V → C be linear. Show that there exists an unique y ∈ V such that
F (x) = hx, yi for all x ∈ V . (8%)
11. Evaluate A100, where A =
−1 −1 −1
1 1 0
−1 −1 0
. (10%)
