89 academic year
Show all works 1. Let U = {(x1, x2, x3, x4) ∈ R4¯
¯ x1+ x2+ x3+ x4 = 0, x1− x2− x3+ 3x4 = 0 } and V be the vector subspace generated by the vector (0, 1, 0, 0) and U .
(a) Find an orthonormal basis of U . (5%)
(b) Find an orthonormal basis of V . (5%)
2. Let x ∈ R, discuss the rank of the matrix
x 0 0 1
1 x 1 0
0 1 x 1
1 0 1 x
. (10%)
3. Let A =
2 2 1 1 3 1 1 2 2
.
(a) Find the characteristic polynomial of A. (5%)
(b) Find the minimal polynomial of A. (5%)
(c) If f (X) = X5− 7X4+ 9X3+ 9X2− 7X + 8, find f (A). (5%) (d) Find an invertible matrix P such that P−1AP is a diagonal matrix. (5%)
4. Define f (x) =
³ Z x
0
e−t2dt
´2
and g(x) = Z 1
0
e−x2(t2+1) t2+ 1 dt.
(a) Show that f0(x) + g0(x) = 0, for all x and deduce that f (x) + g(x) = π
4. (5%) (b) Use (a) to prove that
Z ∞
−∞
e−t2dt =√
π. (5%)
5. Let f be a positive continuous function in [a, b]. Let M be the maximal value of f on [a, b]. Show that lim
n→∞
³ Z b
a
f (x)ndx
´1/n
= M. (10%)
6. Suppose that an> 0, sn= a1+ a2+ · · · + an, and X
an diverges.
(a) Prove that X an
1 + an
diverges. (10%)
(b) What can we say about X an
1 + nan? (10%)
7. Determine all real values of x for which the following series converges:
X∞ n=1
(1 + 1
2 + · · · + 1
n)sin nx n .
(10%) 8. Let (R2, ρ) be a metric space where R2 = {x = (x1, x2) | x1, x2 ∈ R} and
ρ(x, y) = max{|x1− y1|, |x2− y2|}.
Show that the set S = {x ∈ R2|p
x21+ x22 < 1} is an open and connected set. (10%)
