(3) Let {fn} be a sequence of continuous functions on D ⊆ Rp to Rq such that {fn} converges to f on D, and let {xn} be a sequence in D which converges to x ∈ D

90 academic year Part I.

(1) Evaluate the following integrals (15%)

(a) R_∞

0 (^{sin x}_x )²dx. (Hint: R_∞

0 sin x

x dx = ^π₂).

(b) R_∞

0 cos x 1+x² dx.

(2) For sequence {a_n}, a₁ ≤ a₂ ≤ · · · ≤ M for some M ∈ R (15%) (a) Prove lim

n→∞a_n = sup{a_n}, where sup{a_n} denotes the supremum of {a_n}.

(b) Use (a) to prove lim

n→∞(1 + _n¹)ⁿ exists.

(c) Prove the lim

x→∞(1 + ¹_x)^x exists.

(3) Let {fn} be a sequence of continuous functions on D ⊆ R^p to R^q such that {fn} converges to f on D, and let {xn} be a sequence in D which converges to x ∈ D.

Prove or disprove {fn(xn)} converges to f (x). (10%)

(4) Prove C = {x : x = P^∞

n=1 an

3ⁿ, an = 0 or 2} is uncountable. (10%) Part II.

(5)

(a) Give an example of a linear transformation T : R² → R² without any eigenval-

ues. (3%)

(b) Give an example of a linear transformation T : R⁴ → R⁴ without any eigenval-

ues. (3%)

(6) Find the characteristic polynomial, minimal polynomial, eigenvalues and eigenvec- tors of the linear transformation T : R³ → R³ given by

T (x, y, z) = (2x + y, y − z, 2y + 4z)

for all (x, y, z) ∈ R³. Is it possible to diagonalize T ? Why? (6%) (7) Let U₁ and U₂ be vector subspaces of a finite dimensional vector space V . Show

that there is an isomorphism T : V → V such that T maps U1 isomorphically onto

U₂ if and only if dim U₁ = dim U₂. (8%)

(8) Let V be a finite dimensional vector space over R and h·, ·i a symmetric bilinear form on V with hx, xi ≥ 0 for all x ∈ V . Let ϕ: V → V be a linear transformation.

Prove that the following statements are equivalent:

(a) For all x, y ∈ V , hϕ(x), yi = −hx, ϕ(y)i.

(b) The matrix A of ϕ relative to some orthogonal basis of V is anti-symmetric

(i.e., A^T = −A). (10%)

(9) Let V be an n-dimensional vector space over some field F , and let T : V → V be a linear transformation. Let v ∈ V be such that {v, T (v), T²(v), · · ·} spans V . Suppose that dim V = n. Show that the vectors, v, T (v), T²(v), . . . , Tⁿ⁻¹(v) form

a basis of V . (10%)

(10) Let V be a vector space and n ∈ N. Suppose that f : V → V is a linear trans- formation with fⁿ(v) = 0 for all v ∈ V . Let g: V → V be an arbitrary linear transformation and set T = f ◦ g = g ◦ f . Show that there is some m ∈ N such

that T^m(v) = 0 for all v ∈ V . (10%)