M3S2 Spring 2015 - Problem Sheet 4 Solutions
S1) (a)
E(U ) = E
y
V ()
=0
as E(Y ) =
(b)
1
1
2 var (y ) = V ()
(V ()
var(U ) =
as
var(Y ) = V ()
(c)
E
U
=E
1
y
+ V 0 ()
V ()
(V ()2
=
1
V ()
S2) (a)
Z
y
y
1
(y t) dt = yt t2 /2 = (y )2
2
n
X
= D =
(yi i )2
M1M2
Imperial College
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May June 2012
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Mechanics
Date: Thursday. 23 May 2012. Time: 10.00am. Time
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Mathematical Methods H
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M1M2
Mathematical Methods H
Date: examdate Time: examtime
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M1M2
imperial College
London
BBC. MSci and MSc EXAMINATIONS (MATHEMATICS)
May June 2012
This paper is also taken for the relevant examination for the Associateship of the
Royal College of Science.
Mathematical Methods ll
Date: Wednesday, 23 May 2012. Time
IMPERIAL COLLEGE LONDON
Course:
M1A1
Setter:
Professor Andrew Parry
Checker: Dr Daniel Moore
Editor:
Dr Dmitry Turaev
External: Professor Paul Houston
Date:
April 3, 2012
BSc and MSci EXAMINATIONS (MATHEMATICS)
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M1GLA Geometry and Linear Algebra 2014
Solutions Sheet 3
1. (a) The general solution is x = a(1, 1, 1) for any a R.
(b) We spot that (1, 1, 0) is a solution to Ax = b. Thus the general
solution is (1, 1, 0) + a(1, 1, 1).
(c) The system reduces to echelon
M1GLA Geometry and Linear Algebra 2014
Solutions Sheet 9
1. All the claims are direct consequences of defintions.
2. (a) (i) linearly independent, do not span R3
(ii) linearly dependent since v1 + v2 v3 = 0, do not span R3
(iii) linearly independent, span
M1GLA Geometry and Linear Algebra 2014
Solutions Sheet 8
1. Both identities follow straight from the definition of the bracket.
2. The ii-entry of AB is a1i bi1 +. . .+ani bin . The sum of these expressions
over i = 1, . . . , n is symmetric under swappin
M1GLA Geometry and Linear Algebra 2014
Solutions for Sheet 1
1. (i) L = cfw_(1, 2) + (3, 3)| R
(ii) n = (1, 1)
(iii) = 3
(iv) ( 34 , 53 )
2. (i) The line is cfw_a + (b a)| R, which has direction vector b a = (2, 1),
hence normal (1, 2). Hence its equation
M1GLA Geometry and Linear Algebra 2014
Solutions Sheet 5
1. (a) 96; (b) solutions t = 2, 4; (c) x = 0, b c, (a + b + c)/2.
a
2. (i) Eigenvalues are 1 and 3. The corresponding eigenvectors are
a
b
1
1
and
(any non-zero real numbers a, b). So e.g. P =
2b
1
M1GLA Geometry and Linear Algebra 2014
Solutions Sheet 4
1. (i) False for any pair A, B such that AB 6= BA.
0 1
(ii) False, e.g. for A = B =
.
0 0
(iii) True. Assume A1 and B 1 both exist, then
0 = A1 0B 1 = A1 (AB)B 1 = (A1 A)(BB 1 ) = I,
which is visibl
M1GLA Geometry and Linear Algebra 2014
Solutions Sheet 7
1. (i) The line is cfw_(3, 1, 2)+(3, 0, 6)| R. The plane is 2x1 +5x2 x3 =
9.
(ii) The plane is 2x1 x2 + x3 = 1. The line is cfw_(1, 2, 1) +
(2, 1, 1)| R.
2. Write p a = h + u, where h is perpendicul
M1GLA Geometry and Linear Algebra 2014
Solutions Sheet 6
1. (i) Rotate axes (anticlockwise) through/4. If (y1 , y2 ) are
the new
coordinates, then we have x1 = (y1 y2 )/ 2, x2 = (y1 + y2 )/ 2, and
the equation becomes y12 y22 = 4, or y22 /4 y12 /4 = 1, a
M1GLA Geometry and Linear Algebra 2014
Solutions for Sheet 2
1. (a) Unique solution (2, 1, 3, 1).
(b) No solution
(c) General solution (x1 , . . . , x5 ) = (4 a, 10 25 b 3a, 2 + 12 b, b, a) for
any a, b R.
(d) General solution (5a, 2a, a) for any a R.
2.
M1GLA January Test 2014
1
1. (a) Let A =
0
0
3
5
1 2 .
0
1
(i) Find the eigenvalues and eigenvectors of A.
(ii) Giving your reasoning, say whether or not A is diagonalisable.
(iii) Calculate A99 v, where v = (2, 1, 1)T .
(b) For each value of the real num
BSc and MSci EXAMINATIONS (MATHEMATICS)
January 2011
M1GLA (Test)
Geometry & Linear Algebra
Credit will be given for all questions attempted, but extra credit will be given for complete
or nearly complete answers.
The question in Section A will be worth
UNIVERSITY OF LONDON
BSc and MSci
EXAMINATIONS (MATHEMATICS)
May-June2006
This paper is also taken for the relevant examination for the Associateship.
M1GLA
Geometry and Linear Algebra
Date: Friday 12th May 2006
Time: 2 pm 4 pm
Credit will
be given for al
UNIVERSITY OF LONDON
Course: M1GLA
Setter: Skorobogatov
Checker: Liebeck
Editor: Ivanov
External: Cremona
Date:
January 29, 2008
BSc and MSci
EXAMINATIONS (MATHEMATICS)
May-June2007
M1GLA
Geometry and Linear ra
Algeb
Setters signature
.
Checkers signature
UNIVERSITY OF LONDON
Course: M1GLA
Setter: Skorobogatov
Checker: Liebeck
Editor: Ivanov
External: Cremona
Date:
February 5, 2008
BSc and MSci
EXAMINATIONS (MATHEMATICS)
May-June2008
M1GLA
Geometry and Linear ra
Algeb
Setters signature
.
Checkers signature
M1GLA. 2012 January Test Solutions
M1GLA January Test 2011, Solutions
1. (a) x1 + 2x2 = 4 1 Marks)
(b) (1, 2) (1 Marks)
(c) Parabola (2 Marks)
(d) a = 1, b = 3 (2 Marks)
(e) a = 1 (2 Marks)
(f) 1, 1 (1 Marks)
(g) Any matrix of form
a
2a
b
3b
or
a
3a
1
b
,
M1GLA Solutions May 2012
1. (a) The inverse of an n n matrix A is an n n matrix B such that
AB = BA = I. (2 Marks)
(b) To find the inverse of A (if it exists), we need to solve the system
AX = I where X = (xij ) is an n n matrix of unknowns. This system h