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MATHEMATICAL TRIPOS Part IA
Thursday 31 May 2001 9.00 to 12.
PAPER 1
Before you begin read these instructions carefully.
Each question in Section II carries twice the credit of each question in Section I. You may attempt all four questions in Section I and at most five questions from Section II. In Section II no more than three questions on each course may be attempted.
Complete answers are preferred to fragments.
Write on one side of the paper only and begin each answer on a separate sheet.
Write legibly; otherwise you place yourself at a grave disadvantage.
At the end of the examination:
Tie up your answers in two bundles, marked C and D according to the code letter affixed to each question. Attach a blue cover sheet to each bundle; write the code in the box marked ‘SECTION’ on the cover sheet. Do not tie up questions from Section I and Section II in separate bundles.
You must also complete a green master cover sheet listing all the questions attempted by you.
Every cover sheet must bear your examination number and desk number.
SECTION I
1C Algebra and Geometry
Show, using the summation convention or otherwise, that a × (b × c) = (a.c)b − (a.b)c, for a, b, c ∈ R^3.
The function Π : R^3 → R^3 is defined by Π(x) = n × (x × n) where n is a unit vector in R^3. Show that Π is linear and find the elements of a matrix P such that Π(x) = P x for all x ∈ R^3.
Find all solutions to the equation Π(x) = x. Evaluate Π(n). Describe the function Π geometrically. Justify your answer.
2C Algebra and Geometry
Define what is meant by the statement that the vectors x 1 ,... , xn ∈ Rm^ are linearly independent. Determine whether the following vectors x 1 , x 2 , x 3 ∈ R^3 are linearly independent and justify your answer.
x 1 =
(^) , x 2 =
(^) , x 3 =
For the vectors x, y, z taken from a real vector space V consider the statements
A) x, y, z are linearly dependent, B) ∃ α, β, γ ∈ R : αx + βy + γz = 0 , C) ∃ α, β, γ ∈ R, not all = 0 : αx + βy + γz = 0 , D) ∃ α, β ∈ R, not both = 0 : z = αx + βy, E) ∃ α, β ∈ R : z = αx + βy, F) 6 ∃ basis of V that contains all 3 vectors x, y, z.
State if the following implications are true or false (no justification is required):
i) A ⇒ B, vi) B ⇒ A, ii) A ⇒ C, vii) C ⇒ A, iii) A ⇒ D, viii) D ⇒ A, iv) A ⇒ E, ix) E ⇒ A, v) A ⇒ F, x) F ⇒ A.
Paper 1
SECTION II
5C Algebra and Geometry
The matrix
Aα =
1 − 1 2 α + 1 1 α − 1 1 1 + α − 1 α^2 + 4α + 1
defines a linear map Φα : R^3 → R^3 by Φα(x) = Aαx. Find a basis for the kernel of Φα for all values of α ∈ R.
Let B = {b 1 , b 2 , b 3 } and C = {c 1 , c 2 , c 3 } be bases of R^3. Show that there exists a matrix S, to be determined in terms of B and C, such that, for every linear mapping Φ, if Φ has matrix A with respect to B and matrix A′^ with respect to C, then A′^ = S−^1 AS.
For the bases
B =
, C =
find the basis transformation matrix S and calculate S−^1 A 0 S.
Paper 1
6C Algebra and Geometry
Assume that xp is a particular solution to the equation Ax = b with x, b ∈ R^3 and a real 3 × 3 matrix A. Explain why the general solution to Ax = b is given by x = xp + h where h is any vector such that Ah = 0.
Now assume that A is a real symmetric 3×3 matrix with three different eigenvalues λ 1 , λ 2 and λ 3. Show that eigenvectors of A with respect to different eigenvalues are orthogonal. Let xk be a normalised eigenvector of A with respect to the eigenvalue λk, k = 1, 2 , 3. Show that the linear system
(A − λkI)x = b ,
where I denotes the 3 × 3 unit matrix, is solvable if and only if xk.b = 0. Show that the general solution is given by
x =
i 6 =k
b.xi λi−λk xi^ +^ βxk,^ β^ ∈^ R.
[Hint: consider the components of x and b with respect to a basis of eigenvectors of A.]
Consider the matrix A and the vector b
A =
1 2
(^) , b =
Verify that √^13 (1, 1 , 1)T^ and √^12 (1, − 1 , 0)T^ are eigenvectors of A. Show that Ax = b is solvable and find its general solution.
7C Algebra and Geometry
For α, γ ∈ R, α 6 = 0, β ∈ C and ββ > αγ the equation αzz−βz−βz+γ = 0 describes a circle Cαβγ in the complex plane. Find its centre and radius. What does the equation describe if ββ < αγ? Sketch the circles Cαβγ for β = γ = 1 and α = − 2 , − 1 , − 12 , 12 , 1.
Show that the complex function f (z) = βz/β for β 6 = 0 satisfies f (Cαβγ ) = Cαβγ.
[Hint: f (C) = C means that f (z) ∈ C ∀z ∈ C and ∀w ∈ C ∃z ∈ C such that f (z) = w.]
For two circles C 1 and C 2 a function m(C 1 , C 2 ) is defined by
m(C 1 , C 2 ) = max z∈C 1 ,w∈C 2
∣z − w
Prove that m(C 1 , C 2 ) 6 m(C 1 , C 3 ) + m(C 2 , C 3 ). Show that
m(Cα 1 β 1 γ 1 , Cα 2 β 2 γ 2 ) = |α^1 β|α^21 −αα 22 | β^1 |+
β 1 β 1 −α 1 γ 1 |α 1 | +
β 2 β 2 −α 2 γ 2 |α 2 |.
Paper 1 [TURN OVER
11D Analysis I
(i) Show that if g : R → R is twice continuously differentiable then, given > 0, we can find some constant L and δ() > 0 such that
|g(t) − g(α) − g′(α) (t − α)| ≤ L|t − α|^2
for all |t − α| < δ().
(ii) Let f : R → R be twice continuously differentiable on [a, b] (with one-sided derivatives at the end points), let f ′^ and f ′′^ be strictly positive functions and let f (a) < 0 < f (b).
If F (t) = t − (f (t)/f ′(t)) and a sequence {xn} is defined by b = x 0 , xn = F (xn− 1 ) (n > 0), show that x 0 , x 1 , x 2 ,... is a decreasing sequence of points in [a, b] and hence has limit α. What is f (α)? Using part (i) or otherwise estimate the rate of convergence of xn to α, i.e., the behaviour of the absolute value of (xn − α) for large values of n.
12D Analysis I
Explain what it means for a function f : [a, b] → R to be Riemann integrable on [a, b], and give an example of a bounded function that is not Riemann integrable.
Show each of the following statements is true for continuous functions f , but false for general Riemann integrable functions f.
(i) If f : [a, b] → R is such that f (t) ≥ 0 for all t in [a, b] and
∫ (^) b a f^ (t)^ dt^ = 0, then f (t) = 0 for all t in [a, b].
(ii)
∫ (^) t a f^ (x)^ dx^ is differentiable and^
d dt
∫ (^) t a f^ (x)^ dx^ =^ f^ (t).
END OF PAPER
Paper 1
